Aggression Wins Wars

In addition to the political and economic cost of "just" mobilizing (which you could theoretically justify mobilizing without war by saying "look at how our mighty host's assembly caused the enemy to back down!") there is also the military thinking of the era.

Though machineguns/gatling guns, rapid fire weaponry, breech-loading artillery, and smokeless powder (all part of the technological trap that made WWI so deadly) had all been used in war prior to WWI, European thinkers did not fully grasp their import. Wars in America and Asia, fought by "amateurs" and "lesser races" were not seen as definitive. The Franco-Prussian war (which admittedly had some of these new weapons) was seen as much more indicative of what would happen in a general European war. And the takeway from that war was that Speed Kills, and Offensive is All. The hesitant French were knocked onto the strategic defensive, stuck in fortifications, and handily beaten by the Prussians and their allies. Before that there was a similar quick decisive victory of the Prussians over the Austrians, and then before THAT the aggressive nature of Napoleon and other great european/western commanders.

For politicians and military commanders on all sides, there was a firm (and backed by recent historical facts) belief that the side that struck first and hardest would win. So partially mobilizing, waiting for your enemy to mobilize first, was seen as wildly dangerous to the point of it being a military and political nonstarter. And the truth of that is born out in the first few weeks of fighting. The German swing into France would have won any previous war, as would Germany's aggressive move towards the Russians at Tannenburg. The only fail was the Austro-Hungarians, but Conrad Von Hotzendorf is at best the second worst general of the war! It's only in hindsight you can see the massive resilience in million-man conscript armies which enables the war to roll on. Would everyone arming up and sitting at the borders have ended things peacefully? Maybe. But you can't fault them for their thinking because it's the same strategic though that's played out from ancient rome to the Russo-Ukraine war currently ongoing. The only difference is the advent of million-man consript industrial armies meant that the "leadup" was longer so you could watch the enemy make the first move (mobilize) and realize that you either had to beat them to the punch or take an existential risk by letting him get the first blow.

Germany strategically wanted a war

The strategic position of the German Empire prior to WW1 was complicated, with France on one side and Russia on the other.

The general strategy was to hope for a slow Russian mobilization that would allow the Germans to defeat France quickly, thus releasing forces to deal with the big but poorly equipped Russian army.

In this context, mobilizing the army without declaring war would have just given the opportunity to France and Russia to start mobilizing on their own, sacrificing Germany's head start and weakening its position.

Mobilization is hugely expensive

Industry slows down for the lack of workers, agriculture stops for the lack of horses.

It is not something a nation can afford to do lightly.

Speed of mobilization is critical

Germany and France had excellent rail networks, so they mobilized faster. On the other hand, Russia had bigger population resources, so, fully mobilized it would have a larger army (although with worse artillery) - but due to size and relative weakness of railways, it mobilized slower.

Once you draw, you shoot

All the war planners on all sides were blinded by their own excellent technology (conscript armies, magazine rifles, machine guns) and were confident that they were bound to win easily (forgetting that the adversaries had the exact same tech).

Thus it made no sense to them to mobilize (i.e., incur huge expense) and not go to war to justify that expense. Moreover, stopping mobilization "half way" was bound to put the whole transportation and logistical networks in a total disarray, making the country extremely vulnerable to attack in addition to incurring an immense cost.

Thus the German war plans did not even consider "partial mobilization" because they heavily relied on defeating France before Russia could finish its mobilization. The official thinking of the German General Staff was: as soon as Russia started mobilizing (even just "against Austria" as Russia declared) Germany "had no choice" other than to attack France as soon as possible because it stood no chance against both France and Russia fully mobilized (this answers the question in the title).

Note that in the end Germany did lose (despite forcing Russia to surrender), so it does appear reasonable that, if it did not try to take advantage of its shorter mobilization schedule, it would have lost even sooner (please do not consider this sentence "alternative history" ;-)

Prisoner's dilemma

In a way, this is a case of a prisoner's dilemma: all sides would have benefited from cooperation (defusing the situation and avoiding war) but no one trusted the others enough not to mobilize (prepare for the "inevitable defection" of the opponents), which killed all the remaining trust necessary for cooperation.

This, of course, is applicable to the "peace-loving forces" within the major players' governments. The above "cooperation-killing currents" made the war-mongers more powerful.