The Battle of Jutland was not a German defeat. Although the German navy failed to achieve their strategic goal of breaking the power of the British Grand Fleet, they scored a tactical victory, sinking six heavy ships (three battlecruisers, three armored cruisers) for the loss of only two (one battlecruiser, one pre-dreadnaught).

The numeric odds had shifted somewhat against Germany since then, Britain having added seven battleships and two battlecruisers versus Germany's two battleships. Germany planned to counter this by deploying 25 submarines across the Grand Fleet's line of advance, the goal being a larger-scale version of the Action of 19 August 1916, sinking or disabling battleships rather than cruisers.

Although the German commanders expected a repeat of Jutland, inflicting disproportionately high losses despite being outnumbered, the British had remedied the issues that led to their poor performance at Jutland. It's likely that the actual outcome of a rematch would be a disastrous German defeat.

_{Note: The original question title read:}

Q Why German admiralty was so optimistic about the possible outcome of the naval attack on Royal Navy in 1918?

_{This answer still responds to both versions of the title, but parts of the structure are the result of going primarily for the first version of the title.}

"Optimistic" may not be the best word to describe the situation. Neither in the quoted text, nor in all other reality.

Instead of starting with a "why", we might have to start with a 'what'?

What is in the context presented "a favourable result"?

Regardless, since the question title starts with "why", let's do likewise.

The quote presented in the question is from Scheer's book Deutschlands Hochseeflotte im Weltkrieg completed in 1919.

In that book, the full quote gives you an important hint as to 'the why' he in 1919 can give a reasoning for 'a favourable result' at that time in 1918. The date for this deliberation is quite important. The quote presented presents rationales exchanged before 'Naval Order of 24 October 1918'.

For he himself spells it out:

The U-boats liberated from the commercial war materially increased the Fleet's power of attack, and by choosing the point of attack wisely it was highly probable an expedition of the Fleet might achieve a favourable result. If the Fleet suffered losses, it was to be assumed that the enemy's injuries would be in proportion, and that we should still have sufficient forces to protect the U-boat campaign in the North Sea, which would have to be resumed if the negotiations should make imperative a continuation of the struggle with all the means at our disposal.

— Reinhard Scheer: "Germany's High Sea Fleet in the World War", Cassell: London, 1920. (p353, archive.org)

So we see a thinking that re-evaluated the tactical and strategic situation. Clearly seeing an overall unfavourable position, the end near, but still able to act, and hoping to achieve a positive direct outcome for the things at hand.

Crucially, at a point in time that we need to consider as well: it was not the hopefully 'glorious last-stand' with 'all ships going down honourably' that we read about in the quote.

As we read immediately before and after the quoted text

So long as hostilities continued at the front, and there was for the present no indication of their ceasing, the Navy must not remain entirely inactive, while the attacks of the enemy on our Western Front grew ever fiercer, unhindered by any fear of U-boats. A success at sea must have a favourable influence upon the terms of peace, and would help to encourage the people; for the demands of the enemy would depend on the powers of resistance that we were prepared to oppose to them, and upon the consideration whether their own power was sufficiently great to enforce their demands. Anything that would impair their power must be to our advantage.

[Quote in question]

On October 21, when the Note had been dispatched to President Wilson, the U-boats received orders of recall, and my Chief of Staff, Commodore von Levetzow, was commissioned to inform the Fleet Command in Wilhelmshaven of the course of the negotiations, and to take to them the order of the Navy Command : "The forces of the High Sea Fleet are to be made ready for attack and battle with the English Fleet." The Commander-in- Chief of the Fleet, Admiral von Hipper, had already drawn' up plans for such a proceeding, as its necessity was foreseen.

That makes clear that the strategic situation was evaluated as 'peace/armistice negotiations' still ongoing, between October 17 and 19 as presented in the book – before it became known that US President Wilson demanded complete surrender.

Until that outcome became known, such fleet actions were seen as a tactical necessity, perhaps costly to oneself, but worth it on several levels, as increasing the cost for the enemy, supporting the troops on the ground indirectly, and certainly not least as a bargaining chip, in the expected armistice and then peace talks. Weakening the enemy, display of continued willingness and ableness to fight.

A decision nevertheless to go out and then down in glory only came when these hopes were shattered a week later on October 24.

The discussion had to be broken off without result, as the Vice- Chancellor could not be moved to make any concessions. Even when asked if, when the full conditions — in so far as they were tantamount to capitulation — came into force, the people would not be called upon to make a last struggle, Herr Payer only answered "We must first see what the situation would then be."

At an interview the next morning, granted by His Majesty to the Field-Marshal and General Ludendorff, the latter tendered his resignation, which the Emperor accepted.

The Government's answer to Wilson's latest demand was as follows:

"The German Government has duly noted the reply of the President of the United States. The President is aware of the fundamental changes that have taken place and are still taking place in the German Constitution. The peace negotiations will be carried on by a Government of the people, in whose hands the decisive power actually and constitutionally lies. The military forces are also subject to it. The German Government, therefore, looks forward to the proposals for an armistice, which shall lead to a peace of justice, such as the President has defined in his utterances."

The expectation that the negotiations would take a favourable course, as the Government seemed to imagine, was doomed to disappointment. General Ludendorff's prophecy was amply fulfilled; he predicted that if we continued to yield, the end must be disastrous, because the Government had neglected to steel the will of the people for a supreme effort.
But we suffered the bitterest disappointment at the hands of the crews of the Fleet. [From October 29–Novemeber 2, LLC]

Conclusion

The quote describes a line of thinking taking place around October 17, when negotiations with the US under Wilson just became edgier, as the German navy wanted to intensify the U-boat campaign, but from their superiors was forced to turn that down again, with Wilson upping his communicated demands again. A 'favourable result' at that point was seen as almost anything that weakened the allied navies in any way, while keeping German losses in a comparable proportion, so that continued fighting would be seen as still possible in this numbers game for the attrition taking place. No 'decisive win', 'no glorious going down' either. But an overly hopeful tactical decision to get better terms in the peace talks while maintaining morale and defense capabilities as much as possible. That then indeed all changed, radicalized, a few days later to eventually accept all losses that might – or quite certainly would – have been amassed.

Extra emphasis note: these are about Scheer's memoirs. This answer so far has explained his view as presented by himself in 1919. As such, they are tainted. Historical research shows that this kind of 'limited at first' reactivity is most probably to a degree a delusion itself, albeit one that Scheer and his friends from the Seekriegsleitung did present indeed in a similar manner – that is distorted and not in full – to the Reichsleitung at the time.

But historical research concludes that already from October 8 onwards Scheer and the Seekriegsleitung were quite firmly more on course to what then was issued on October 24 then what he presents in the quoted passage. What Scheer conveniently leaves out of his picture is the actual 'win-or-sink order' of October 24 itself! In his version, the mutineering Bolshevik sailors could not see the wisdom of the plans of their superiors, with 'acceptable but necessary losses', which we see downplayed in his memoirs with our topic here. Plus in fact, the admirals were those who started a rebellion against superiors, as emperor and chancellor had expressly forbidden actually doing anything that would resemble the Operationsbefehl Nr 19. The emperor even insisted on getting approval for such things from the cabinet would be mandatory.

Compare: "Überlegungen in ernster Stunde" (Considerations in a serious hour, excerpts quoted here), a concept for the renewed deployment of the high seas fleet, drafted by Admiral v. Trotha, endorsed by Admiral v. Hipper, October 8, 1918:

• 1. even now it must precede all other considerations: "How can submarine warfare be maintained in strongest effect." […]
• 3. thus the fleet is bound by the submarine war; an advance of the entire high seas forces to seek success on the water […] would mean abandoning the basis for the submarine war.
• 4. Such an operation is therefore possible only: a. if the enemy breaks into the German Bight or the Belts. b. if submarine warfare is completely abandoned. c. if serious damage to British naval power promises more advantage to us than the continuation of the submarine war, or d. our fleet otherwise meets an ignominious end.
• 5. such a final battle is the ultimate goal of the fleet, so that we do not have to conclude this war without the national strength it contains having been brought to bear to the fullest extent.
     From an honorable fight of the fleet, even if it becomes a death struggle in this war, a new German fleet of the future will emerge - if our people do not fail nationally in general; a fleet bound by disgraceful peace has its future broken. […]

As is clear from Scheer's quote about the changed tactical situation: they just tried to intensify submarine warfare, but were directly ordered to call that off. Thus that entire force became available for an attack on the Royal Navy itself.

Bullet points 4 and 5 however also demonstrate that "go down in glory" or 'let's die like samurai' is not the only/unavoidable outcome desired. That was but one outcome, although fully anticipated, albeit a rather likely one. Item 4c draws on the possibility that a full battle sortie might end up similar to Jutland/Skagerak, in dealing a lot of damage to the enemy, without going down entirely, but with just 'you should look at the other guy'.

Item 4d however really means: 'if we do not engage in full effort now, it would be very shameful for us'. That is directly leading to item 5 in which the dreaded word peace is avoided in favour of "future", which in turn really stands in for 'the next war' and the connected political developments.

These deliberations are directly from the inside, and whether "go down with honour" really is the main or even only point to observe is quite a bit doubtful for October 8, but quite obviously these aspects of that plan seem to have come to dominate the mood when the final order was given out. So much so that the sailors saw only that aspect and rebelled. Curiously, many later historians also over-focus on this death-cult aspect of a giant Ragnarök moment, often assuming out of hindsight a stringency and inevitability, neglecting the dynamics of the process, the information available to the planners, their real hopes, of the remote possibility that such an Endkampf – the final struggle – might even turn the tide, however ill founded those might have been.

These deliberations were also strictly internal, and no mention was made to either emperor or chancellor in meetings supposed to discuss such things on October 18/20. It was expected that the emperor would not be pleased and the chancellor was 'not supposed to have any say in such matters anyway'. As such another "favourable outcome", not mentioned by Scheer, but often analysed as to be a very probable case: no matter what would have happened at sea, one calculation made was: a win would topple the new government, and a loss would topple it as well. Which might read a bit cynical.

— Ulrich Kluge: "Soldatenräte und Revolution. Studien zur Militärpolitik in Deutschland 1918/19", Vandenhoek & Ruprecht: Göttingen, 1975, p371. link
— Michael Epkenhans: "Meuterei der Matrosen 1918 – Fanal der Revolution und Trauma der Marine", Presentation, Zentrum für Militärgeschichte und Sozialwissenschaften der Bundeswehr in Potsdam, 2018 (PDF).
— Wilhelm Deist: "Die Politik der Seekriegsleitung und die Rebellion der Flotte Ende Oktober 1918", Vierteljahreshefte für Zeitgeschichte, No 4, Vol 14, 1966 (PDF).
— Leonidas E.Hill: "Signal Zur Konterrevolution? Der Plan zum letzten Vorstoß der deutschen Hochseeflotte am 30. Oktober 1918", Vierteljahreshefte für Zeitgeschichte, No 1, Vol 33, 1988, (PDF).
— Holger Afflerbach: "„Mit Wehender Fahne Untergehen” Kapitulationsverweigerungen in der deutschen Marine", Vierteljahreshefte für Zeitgeschichte, No 1, Vol 49, 2001 (PDF).

By mid-October 1918, it appeared to many contemporaries, especially among conservative and nationally orientated military and political elites, that the German government was close to rejecting the exchange of notes and proclaiming Endkampf. Notably, on 17 October, while a handful of naval planners were secretly working on the operation, Scheer and Levetzow attended a meeting between Ludendorff and the cabinet of Prince Max von Baden. Its main purpose was to deliberate the German response to the second American note, a note that liberal State Secretary Conrad Haußmann, a member of Prince Max’s cabinet, described as ‘exploding like a bomb’ when its contents, including the stipulation that the Entente would control the speed of German military withdrawal from occupied France and Belgium, were made public in Germany. By the end of the meeting even Ludendorff openly spoke of Endkampf’s prospects for success. In turn, after the German government responded to the Americans with a further conciliatory note on 20 October, four days later, on 24 October, the same day the planning stage of the naval operation was declared complete, the OHL, the organization that demanded a ceasefire at the end of September, now attempted to publicly sabotage the work of Prince Max’s government and end the diplomatic process. On this occasion, Ludendorff was overruled. He was summoned to Berlin and dismissed on 26 October 1918. This was the point when the navy’s commanders thought their hour had arrived. In their eyes either the German surface fleet would justify its existence in battle and remobilize the German public to continue fighting; or, according to the logic of their ultra-nationalist worldview, fleet and nation would fail to exist.

The officers’ decision making is often represented as something peculiar to a group of men whose expectations had been defined by the unique world of the Imperial German Navy’s officers’ corps. So too, some historians have dismissed discourses calling for Endkampf as nothing more than ‘rousing calls for perseverance’ and a ‘final propaganda crusade’. And yet, the more we think about how German military planning interacted with a much broader political and social conversation about Endkampf, the more it becomes clear that the prospect of Endkampf was one of the most important aspects of German strategic thought during the weeks leading up to the Armistices of November 1918. The naval command was not unique: it was only one of a significant number of organizations thinking about how Endkampf could be realized.

— Mark Jones: "Founding Weimar. Violence and the German Revolution of 1918–1919", ch "La Grande Peur of November 1918", Cambridge University Press: Cabridge, New York, 2016.