Short answer: You've just witnessed how your country went from a democracy to fascism in just a few years, murdered millions of its own citizens and killed many more millions across the continent. Now you've been given the task of writing a new constitution. What do you do?
Given the then very recent atrocities, the first thing you write down is that Human dignity shall be inviolable. To respect and protect it shall be the duty of all state authority. With that out of the way, you analyze what went wrong with the old constitution, and decide the new constitution must be resistant against attempts to abolish it or alter its basic principles. Part of that is that some articles can't have their essential meaning changed, others can't be changed at all. But it goes further -- organizations that have the goal to abolish the constitution can be banned (along with symbols that represent them), and if the Federal Constitutional Court agrees, even parties (so far a successor party of the NSDAP and a communist party in the 50s.) Also, no speech that is capable of inciting violence against minorities.
Those are the limits. I realize some of the consequences sound ridiculous to Americans, but you have to see it in the historical context. Also, some impressions Americans often have about those limitations are simply not true. For example, showing swastikas. I've seen plenty of swastikas in history class or in movies. That's perfectly legal (education/art). A T-shirt with the NSDAP flag, on the other hand, can indeed get you a fine of several hundred euros.
In practice, Freedom of Expression is alive and well in Germany (unless you're a Nazi.) There are no beeps during TV shows and wardrobe malfunctions are something to laugh about. You're much less likely to get sued, and civil and criminal sentences are much lower (incarceration rate is a bit over a tenth of the US'.) Nobody raises an eyebrow when you proclaim that you're an Atheist and several openly gay politicians have been elected into high offices (two equivalent to a governor and our new Foreign Minister/Vice Chancellor, for example.)
When you read that the president can't veto a law, keep in mind that he's merely the Head of State. The US President is also the Head of Government, and elected directly (in practice.) Sufficient to say that last time those positions were held by the same person, it didn't work out that well for us. The parliament elects the chancellor, and the parliament can also elect a new one at any time. The ability to get rid of a Head of Government, without an "impeachable offense", can be useful at times. The parliament has proportional representation (with the limitation that only parties that get >5% of the votes are taken into account), so it almost never happens that a single party can form the government on its own, and those coalitions can break apart to form a new government with other parties. Finally, the courts usually do a very good job, some attempts to introduce particularly stupid laws you may have heard about backfire and we get a new Fundamental Right out of it.
One more thing that may be important: Election campaigns, particularly financing, work differently. Parties and their candidates get most of their financing out of tax money, depending on how many votes they had in the last elections, and membership fees. There's a limited number of campaign spot slots available that get assigned the same way, you can't just buy more. Also, no PACs.
