I first want to give you a map of the general region we're talking about in 1914. It's focused on the end of 1914 after the front began to stabilize but we can still look at some important landmarks and get an idea of what we're talking about here.

Let's observe some notably landmarks. Luxemburg sits comfortably as that little blob attached to France, Belgium, and Germany. As you follow that line up through the Ardennes Forest (squared in purple) we are running along the shared German/Belgian border. At the top where it diverges into two paths, that is where the Belgian/Dutch border begins. The shared border of Belgium and Germany would be 80 kilometers, not a lot of room to maneuver through as you're already imagining -- especially with that huge, dense forest covering up like half of it.

Liege is a critical fort, squared in purple likewise, stationed between two major rivers -- notably the Meuse coming from the South. Controlling Liege means controlling the proverbial keys to the Belgian countryside, it was easy access across that mammoth river and toward Brussels -- their capital -- and toward that aforementioned open country where the 1st - 3rd Armies could have room to operate. This was a critical path to open up because the main component of the German war plan, Alexander von Kluck's First Army of 120 battalions -- massively reinforced to 320,000 men -- and 748 guns, would be tasked to rush toward the English Channel and then 'hug the coast' and envelop the French. Karl von Bulow's Second Army, of 137 battalions -- 260,000 men -- and 820 guns, would be tasked with a similar duty. It would be the force that would fight in infamous battles at Mons and Charleroi (near that red circle on the map, just North) and secure the center of Belgium and be the primary driving force into France.

Both of these jobs would require immediate advancement past the Meuse river however, and Liege was the closest place to the border to attack with the best infrastructure to support 257 battalions of men -- about 580,000 men -- moving through it at the same time to swarm into the Belgian countryside and advance into France. Meanwhile Hausen's Third Army of 101 battalions and 596 guns would be called the Schwenkungsflugel -- translated to 'swing arm' and of a meager 180,000 men -- would advance down through the Ardennes, to Dinant (on the Meuse down there) and cross there after securing Southeastern Belgium. Their job would be then to assist Bulow's Second Army as needed or to redirect South to the Duke Albrecht's Fourth Army in support in the Luxemburg region.

Now let's get past all the fancy numbers and statistics for a second and really analyze what's going on and get to the meat of your question -- why the heck didn't the Germans just go through the Netherlands? There were forts there certainly, but not nearly at the level of Belgium and it would have allowed for a massively expanded front which would have spread out all those men and let them swing more naturally into Belgium and North France. It's likely it would have sped things up by a couple of days even and would not have required for such a desperate assault on Liege and later Dinant. Schlieffen himself actually saw no issue with this and included it in his original 1905 plan. However, Helmuth von Moltke saw issue with that. He recognized in the 1910's that Britain would most likely become involved on the side of France after a breach of Belgian neutrality and they would perform a full blockade of Germany. They would have absolutely no way of dealing with the blockade, their navy was simply not capable of contesting it. That's where the Netherlands comes in -- by not invading them and keeping them as a neutral border state Germany can trade for vital goods through the Netherlands who will retain full trading capacity with the rest of the world. It was sort of a back up plan in case things didn't go as well instead of going "all in".

Okay, so we can't just avoid Liege (which was really Germany's only issue with the Belgian military as an organized force) -- that has to be attacked. What about the French forts that they all attacked at Dunkirk and Maubeuge (that red circle you see on the map that held out for a long ass time)? Well, you're kind of forced to deal with these forts as you pass them. They were attacked for the same reason any army attacked any fort in the history of the world instead of just bypassing it and continuing on with the offensive -- it holds tens of thousands of men. In the case of Maubeuge almost 50,000 men and with similar stories in other French forts. If you just walk past them those men are now behind your lines and can cause havoc. It is a projection of force in that region and if not dealt with allows the enemy to perform military operations with a place to fall back on safely and disrupts the entire offensive. It has to be dealt with or you just get pockets of 50,000 men popping up everywhere in critical areas like rivers crossings and major cities all behind your lines.

This is actually precisely how the Germans would throw away their successful offensive. In early September the Germans would be on the outskirts of Paris, also squared in purple, and had the entire French army on its heels. Or rather, "on its heels." They had an option then -- siege the fortress of Paris or continue to pursue the French armies, push them further Southeast into the 4th-6th armies sitting on the French/German border and completely encircle everyone. The latter would require basically ignoring Paris and considering even minor forts took a significant amount of time to subdue and it would take days if not weeks to move the super heavy 205mm artillery necessary to even think about sieging Paris, the Germans opted to completely ignore it and continue pursuing the French. This was a horrible option. The Parisian forces would notice that Bulow's Second Army had given them their back as they turned South and Kluck's First Army was separated from the rest of the pack just North of them too. They, along with the entirety of the British Expeditionary Force, would pour out of and around Paris into this gap and nearly encircle both the 1st and 2nd German armies. This is why they sieged all those forts before and it's why people have been sieging forts since the beginning of time.

I hope that cleared some things up left out by Mr. Carlin ;)

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Notes:

"The Marne: The Opening of World War I", Holger Herwig

"The First World War: Volume I: To Arms", Hew Strachan

"The First World War: Germany and Austria-Hungar​y 1914-1918", Holger Herwig

(As you can tell, Holger Herwig is my boy. He's a really good, but extremely dense and in depth source. Lots of army names, numbers, officer names, and so forth. Not the sexiest thing out there but it's pretty helpful in discussions like this)