Not Austria, Prussia received the 'best share' in the end.

It's not about size. Neither population nor territorial size. It's purely location, location, location. In terms of strategic positioning.

It was most strategically important for Prussia itself, and a long standing goal, not just a 'border correction', but arrondis*m*nt and strategic importance. For Prussia internally and by cutting off Poland-Lithuania from the sea also usurping large percentage of the external trade, with ample tarriffs, thus weakening it significantly.
While Prussia gained much from this deal, and Russia expanding also elsewhere and 'threatening Austria' with it, both sides were already near closing a deal that now had to be sold to Austria as 'a sweet one', by offering Austria its share of the spoils. Therefore, if we indulge ourselves in using this value judgement of 'best', at all, Austria was late to that game, the weakest but not powerless part, and practically bribed and pressured into agreement, which within Austria was not 'protested' much, but mostly favoured anyway, with the most important exception of Maria Thersia herself.

Statistics, unimportant

The sheer numbers often presented in terms of territory don't tell the proper story:

    Prussia           Russia          Austria        Total
  km²     pop.     km²     pop.     km²     pop.       km²     pop. 
34900   356000   84000  1256000   83900  2669000   202.800  4281000

The state of affairs

Russia and Austria initially rejected the idea of annexation of Polish territory in principle, but they warmed up to it. The decisive leitmotif was maintaining a balance of power.

After Russia had advanced at the expense of the Ottoman Empire just recently and a more general Russian expansion in south and eastern Europe was on the horizon, both the Hohenzollern and Habsburgs felt a bit threatened. This seemed ever more real when Russia invaded Poland to quell the sometimes so called 'first Polish uprising', as the civil war and the resulting Russo-Polish-war that resulted from the anti-Russian Confederation of Bar unfolded, from 1768–1772, the last being also the year of the partition.

Prussia's and Austria's opposition to such unilateral territorial gains and the associated increase in Russian power gave rise to plans for all-round territorial compensation go-around.

[…] the Sultan declared war on Russia in October 1768. All of a sudden, two originally separate conflict zones - Poland and the Balkans - were suddenly connected in an explosive way. The question was how the Habsburg Empire should react to this. The old ally Russia had become a threat; conversely, since the end of the last war in 1739 the former enemies, the Turks, had transformed themselves into reliable, peaceful neighbours. Poland-Lithuania was an important buffer against the Tsarist Empire. Maria Theresa did not want to jeopardise this stability. Kaunitz, on the other hand, saw great opportunities opening up. As early as 1768 he drew up one of his notoriously far-reaching plans for exchange and partition: Frederick II was to return Silesia to the Habsburgs and be compensated for this with Polish territory. It did not come to that, the plan still seemed too chimerical at that time. But the idea of coming to terms with the old enemy Prussia and using the weakened Poland as a disposable asset was in the world. In order to counter the Russian urge for expansion and at the same time to capitalize on the war against the Ottoman Empire, Kaunitz considered it advisable to move closer to Prussia. (Barbara Stollberg-Rilinger)

In diplomacy, initiative favours the bold

The most important: Frederick II now saw the opportunity to realize his expansionist plans and increased his diplomatic offensive. He referred to a proposal already explored in 1769, the so-called "Lynar Project", a ruse with even a 'false name' he sold as an ideal way out to avoid a shift in the balance of power:
Russia should please Austria's interest, now by renouncing the occupation of the principalities of Moldavia and Wallachia. Since Russia would not agree to this without compensation, they were to be offered a territorial equivalent in the east of Poland as 'a compromise'. At the same time, Prussia was to be given Russian support and namely for acquiring the territories on the Baltic Sea which it was seeking all along.
So that Austria would also agree to such a plan, the Habsburg Monarchy was finally to be given the Galician parts of Poland as well. Generosity is easy if you give away things you do not own.

So while Hohenzollern policy simply continued to aim at the consolidation of West Prussian territory, Austria was offered the chance of a small compensation for the previous loss of Silesia to Prussia in 1740 (cf. Silesian Wars).

However, it would have been an unforgivable political mistake to blindly rely on the honesty of the Austrians. In the circumstances of the time, however, when Russia's preponderance was becoming too great and it was impossible to foresee what limits it would place on its conquests, it was very appropriate to approach the Viennese court. Prussia still felt the blows that Russia had dealt him in the last war. It was by no means in the king's interest to work on the expansion of such a fearsome and dangerous power himself.

The choice was either to stop Russia in the course of her mighty conquests, or, which was wiser, to make skilful use of it. The king had not failed in this respect. He had sent to Petersburg a political project which he attributed to Count Lynar, known from the last wars for having brought about the Zeven Convention between the Hanoverians camped near Stade under the Duke of Cumberland and the French under the Duke of Richelieu1. The project contained a sketch of a division of some Polish provinces between Russia, Austria and Prussia. The advantage of this division was that Russia could calmly continue its war against the Turks without fear of being hampered in its undertakings by a diversion which the Empress Queen could have easily made by sending a corps of troops to the Dniester, for this would have cut off the Russian armies from Poland, from which they obtained most of their food.

Lynar's project, designed by the king himself, was contained in a decree to Count Solms of 2 February 1769. The decree reads: "Count Lynar has come to Berlin to marry his daughter to the son of Count Kamel. It lsi the same one who concluded the convention of Kloster Zeven. He is a great politician and still guides Europe from his village, where he has retired (Lübbenau in the Spreewald). This Count Lynar has fallen for something quite strange, in order to unite all the interests of the princes in favour of Russia and to give the European affairs of state a completely different turn at a stroke. According to his plan, Russia is to offer the Viennese court the city of Lviv and its surroundings and the Spiš region in order to help it against the Turks. Russia itself is to acquire any piece of Poland as compensation for its wartime expenses, and since all jealousy between Austria and Prussia will then have been eliminated, both of them will stand by the Russians in a race against the Turks. There is something dazzling and bribing about this plan. I felt I had to tell you about it. You know how Count Panin thinks, so you will either keep it quiet or use it as you see fit... ...but it seems to me more brilliant than certain."
_{— Œuvres de Frédéric le Grand - Werke Friedrichs des Großen Digitale Ausgabe der Universitätsbibliothek Trier. Die Werke Friedrichs des Großen : in deutscher Übersetzung; 5 Denkwürdigkeiten vom Hubertusburger Frieden bis zum Ende der Polnischen Teilung 1. Kapitel. Die Politik von 1763 bis 1774.}

Habsburg's 'timidity' explained

However, as correctly pointed out already, Maria Theresa, according to her own statements, had "moral reservations" and was reluctant to allow her claims for compensation to take effect at the expense of an "innocent third party" and, what is more, a Catholic state. She herself concluded that from now on "good faith is lost for all time", which was described as "she cried, but she took". Her lamentations may have been in part sincere, but in the end it looks like a case of crocodile tears.

Precedence had set the stage

Yet it was precisely the Habsburg Monarchy which in 1769 prejudiced such a division by "reincorporating" by force 13 towns or market towns and 275 villages in the Spiš county (Gespanschaft/Župa Zips). Not just a lofty plan for divisions by Kaunitz but actively taking territory by force and 'against the rules'. These villages had been ceded by the Kingdom of Hungary to Poland in 1412 by way of a pledge/collateral and simply had not been redeemed later.

According to at least one historian it was this military action that initiated or at least inspired the actual partition action now unfolding (Georg Holmsten: "Friedrich II. in Selbstzeugnissen und Bilddokumenten", Rowohlt: Reinbek bei Hamburg 1979, p146.). Be it as precedence, and demonstration of weakness on Poland's part, and willingness of the surrounding great powers to let such things happen.

Prince Heinrich reports on January 8, 1771, about his conversation with the King (Frederick) that he had been with Empress Catherine that evening: She jokingly told me that the Austrians had seized two Starostei in Poland and had planted their border eagles on the borders of these territories. She added: "But why shouldn't all the world also seize them? I replied that you, dear brother, had drawn a border cordon (against the plague) in Poland, but had not occupied any Starostei. "But why not occupy?" said the empress laughing. A moment later Count Chernyshev approached me, brought the conversation to the same subject and concluded: "But why not take away the diocese of Warmia (Ermland)? "After all, everyone must have something."
_{— Œuvres de Frédéric le Grand - Werke Friedrichs des Großen Edition numérisée par la Bibliothèque Universitaire de Trèves. Die Werke Friedrichs des Großen : in deutscher Übersetzung; 5 Denkwürdigkeiten vom Hubertusburger Frieden bis zum Ende der Polnischen Teilung — 1. Kapitel. Die Politik von 1763 bis 1774 (Explanations in brackets added, LLC)}

While Maria-Theresia was still in consultation with her son Joseph II – who was in favour of a partition – and State Chancellor Wenceslas Anton Kaunitz, Prussia and Russia already concluded a separate partition agreement as early as 17 February 1772, thus put Austria under pressure to act.

Austria's allure and advantages accepted

In the end, the monarch's concern about a shift or even a loss of power and influence outweighed the risk of an antagonism with the two powers. The Polish territory was not to be divided among them alone, und thus Austria joined the partition treaty. Although there was some hesitation on Habsburg's side, there had already been attempts by the State Chancellor of Kaunitz at the end of the 1760s to conclude an exchange deal with Prussia, in which Austria was to get Silesia back and in return support Prussia in its plans to consolidate Polish Prussia, the prime objective for Prussia now as well, and again.

Austria was thus almost lured into taking an active part in the partition. The Russian plans suited them in view of the plans that had been circulating for years before and provided a welcome opportunity to implement their own interests.

Focussing on Maria Theresia's views:

Throughout the entire period, the empress stubbornly resisted any kind of partage at the expense of third parties. Her aim was to "get out of the tangled web of things with equity", even without any personal advantage. She feared that the further advance of her troops in Poland would make Prussia, Russia and Turkey her enemies. Instead, she wanted to engage with all three powers in a peace agreement that would not entail any major territorial renvers*m*nts. However, she was also not prepared to accept that Russia and Prussia would gain advantages in Poland: "In that case, [I] could not go away empty-handed. But there was no question of enriching oneself at the expense of Poland, just as there was no question of enriching oneself at the expense of Turkey, with which it had just concluded an agreement in good faith. The only one who could be considered for compensation was the King of Prussia, for example with the county of Glatz or with one of its western territories. "By this clear and sincere performance I believe, and I alone find, that we could still come out of this with honor, and perhaps with some nuzz or even less for the balance." Her letters to Joseph and Kaunitz left nothing to be desired in terms of clarity.

In January 1772, she described the whole policy on the Russian-Turkish question as misguided: the deployment of Italian and Dutch troops in Hungary, the unfortunate agreement with the Turks, the threatening tone of voice against the Russians, the mysterious attitude towards enemies and allies, and in general this whole tendency to use the war between Russians and Turks as a means of extending one's own borders, giving oneself the appearance of respectability, honnêteté. She called it "politics à la prussienne." Even if Kaunitz's diplomacy should succeed in winning over Wallachia or even Belgrade, she found the moral price for this too high: "Throughout my entire unfortunate reign we have at least tried to maintain an honest and decent attitude of honesty, moderation and reliability in our commitments in everything. This has earned us the trust, even dare I say the admiration of Europe and the respect of our enemies. It has all been gone for a year. […] Nothing in the world pains me more than the loss of our good reputation. Sadly, however, I have to admit that we deserve it, and that is what I want to change by rejecting the evil and ruinous principle of profiting from the conflicts of others […]".

In a long, fundamental memorandum dated February 5, 1772, she once again made her view of things clear with all her insistence. Prussia, not Russia, was and remained her most threatening enemy and its enlargement the greatest evil. Instead of acting as an impartial mediator between Russians and Turks, earning the thanks of both powers and preventing the dismemberment of Poland, they had thrown themselves in the greatest blindness at the Prussian king. By occupying Polish territory they would have given him the pretext to do the same. By a series of "wrong, ill-calculated, inconsistent and dangerous steps" they would have manoeuvred themselves into the embarrassing position of having to contribute to the enlargement of two powers, our rivals and enemies, and in exchange, so to speak as a gift, to receive from them something to which they have no more right of disposal than we have a right of acquisition. By what right can one deprive an innocent man of the right to defend and support the one we have always prided ourselves on?

The argument of convenience - not being able to stand aside when others get rich - did not hold water. Among private individuals, this is what is called shame and injustice - should the laws of natural law not apply to rulers at all? Shortly afterwards, when Prussia and Russia had formally agreed to annex Polish territories, she wrote to Kaunitz: "All parties are unfair in their reason and harmful to us. I cannot regret this request enough and must confess that I am ashamed to be seen." That her view of things not only had morality on her side, but was also politically sensible, can hardly be denied.

Meanwhile, the Austrian envoy in Berlin pushed up Vienna's compensation demands. In the end the empress submitted to the policies of her son and her chancellor of state. Despite all her scruples and objections, she did not prevent the formal partition agreement between the three powers from being concluded in St. Petersburg on 5 August 1772. She was the one who signed it in the end. Shortly before that, on August 3, she had resignedly written to Kaunitz: "I find that there is nothing else to do before now, but I cannot yet be reassured about the increase of these 'double puissancen' and even less that we should also part with the same". She and her daughter Marie Antoinette, now wife of the French heir to the throne, were then left to appease her grandfather-in-law Louis XV, a traditional friend of Poland. For France, like England, had been completely left out of the whole affair.
_{— Barbara Stollberg-Rilinger: "Maria Theresia : die Kaiserin in ihrer Zeit: eine Biographie", CH Beck: München, 2017. (p560–580)}

Outcomes for winners, and Poland

The partition treaty between Prussia, Russia and Austria was declared to be a "measure" for the "pacification" of Poland and meant the loss of over a third of its population and over a quarter of its previous territory for Poland, including the economically so important access to the Baltic Sea with the mouth of the Vistula.

Prussia gained what it had been striving for so long: with the exception of the cities of Danzig and Thorn, the entire territory of the Prussian Royal Share and the so-called Netzedistrikt were given to the Hohenzollern monarchy.
In terms of size and population it thus received the smallest share. Strategically, however, it acquired the most important territory and thus profited considerably and perhaps by far the most from the First Partition of Poland.

An important desideratum of territorial-state and dynastic prestige was fulfilled. In the future West Prussia was to form the indispensable 'tendon' of Prussia in the northeast, both strategically and economically.
_{— Martin Broszat: "200 Jahre deutsche Polenpolitik", Suhrkamp: Frankfurt a. M., 1986. p50}

Furthermore: in the future, the Hohenzollern king was thus allowed to call himself "King of Prussia" and not only "King in Prussia", a not so small but quite important status upgrade. Russia renounced the Danube principalities Moldavia and Wallachia, but was instead granted the territory of Polish Livonia and the Belarusian territories up to the Daugava. Austria secured the Galician territory with the city of Lemberg/Lwów/Lviv as its centre with parts of Małopolska ('Lesser Poland').

With this Poland, the once largest territorial state in Europe after Russia, now became the crippled plaything of its neighbours. And Frederick II himself described the partition of Poland in 1779 as an outstanding success of novel crisis management.
_{(— Michael G. Müller: "Die Teilungen Polens 1772, 1793, 1795", Beck: München, 1984.)}