Why did Germany attack Neutral Belgium in the first world war?

Same reason the chicken crossed the road, to get to the other side.

More seriously, the Schlieffen Plan depended on knocking out France before Russia completed its mobilization. Since Russia had publicly begun mobilization before the war, as a way of showing Germany its seriousness, Germany felt extra time pressure to get the job on with France. It seemed like a risk-reward thing, even though invading Belgium assured Britain's commitment to the war and the behavior of German troops in Belgium turned (Western) world public opinion against Germany.

Another way to look at it is that there were two hands in Germany and one hand did not talk to the other. The political/diplomatic hand had gone through July 1914 making a bet that they could force Russia to back down from its defense of Serbia, isolate it from France, and otherwise create a balance of power that favor Germany. The Schlieffen Plan was not part of the consideration.

When that gamble failed, Germany had to put its money where its mouth was, or face the same loss of prestige that they had tried to force from Russia. So the buck passed to the other hand, the military wing, which was now tasked with fighting a two-front war. Now the General Staff had great prestige, having won three remarkable campaigns in the 1860s to 1871, along with some impressive colonial war action in the 1890s to 1911. They said, "we need to go through the Netherlands and Belgium." The Kaiser took the Netherlands off the table. That at least going through Belgium was necessary in order to assure the timely destruction of the enemy in France was not a matter of great doubt.

I recommend Isabel Hull's "Absolute Destruction: Military Culture and the Practices of War in Imperial Germany".

It may seem absurd to us today, but in 1914 the German General Staff fully believed that it was possible to march through Belgium and simultaneously maintain her neutrality. The Schlieffen Plan was in fact premised on this assumption.

You rightly ask how this absurd notion arose - in Napoleon's 1805 campaign when the French marched through Ansbach-Bayreuth, then Prussian territory, en route to surrounding Mack's forces at Ulm. Napoleon steadily maintained that such march did not violate Prussian neutrality because the French troops were merely in transit. While the Prussian's blustered at this, and a year later cited it as a casus belli, none-the-less Napoleon's dramatic victory at Austerlitz resulted in them, temporarily at least, accepting Napoleon's argument. (Note the line of march to Ingolstadt for the corps of Marmont, Bernadotte, and Deroi.)

Having been the victim of this ruse in one war against the French, they thought it was fair game to employ it themselves, tit-for-tat, in another.

This was of course naïve in the extreme. Belgian Neutrality was explicitly guaranteed by the Treaty of London (1839) of which Prussia, as a member of the German Confederation, was a signatory. The intervening eleven decades had dramatically altered the political dimensions of warfare due to the emergence of Nationalism in the latter stages of the Napoleonic era. The atrocities committed by German soldiers against a Belgian populace outraged by the violation of their neutrality quickly turned worldwide popular opinion against the Germans. And the troops quickly assigned to protect supply lines through Belgium were first-line troops diverted from the Schlieffen Plan's big right wing, and were sorely missed on The Marne.

An important lesson to draw from this is the need for civilian oversight of military planners in our age. No longer can military expediency be regarded as sufficient motivation for military action. While the Prussian (and then German) General Staff was far and away the most capable in the world technically through the decades from 1840 to 1945, it's failure to accept or even recognize the need for civilian oversight post-Bismarck was repeatedly it's undoing.

The Germans in fact made such a request. Of course the word "request" in this context should really have irony quotes, considering the disparity in power between the two countries, and the fact that Germany was not going to take "no" for an answer.

Germany wanted free escort through Belgium (and originally the Netherlands as well, which plan Kaiser Wilhelm II rejected) to invade France. Neutral Belgium rejected this idea, so the Germans decided to invade through Belgium instead. France also wanted to move their troops into Belgium, but Belgium originally rejected this "suggestion" as well, in the hope of avoiding any war on Belgian soil.

So bascially the Germans attacked because attacking France through Belgian territory was The Plan, and the Belgians refused to just let them walk through unopposed.