I agree with other responses that the first two questions are: (i) Can you afford it? and, (ii) Why the US?
One issue is that university in the US is expensive. A small handful of universities in the US offer need based or in some cases merit based aid for international students. A small handful offer sports scholarships if you are exceptional at a sport that they care about (a few Canadians for example play hockey for a few universities in the US – a few Europeans most likely play the sport we call soccer where you kick the ball with your foot). Unless you somehow get very good financial aid, the cost of university in the US is very high. $80,000 per year for four years is not unusual. Many (probably most) students in the US attend their in-state public universities because it is more affordable, but as an international student you do not have one (at least not in the US).
Another issue is that once you graduate from university in the US, as an international student you will most likely be required to leave. When you apply to university in the US, if you tell immigration “I would like to study in the US because I want to stay in the US after graduation” then you are likely to be rejected for the student visa and not be allowed to come here to study. Whether having a degree from a university in the US will help or harm your chances of getting a job back in your home country will depend upon where you come from. I do not know the situation in Germany.
The most famous and best known universities in the US are indeed very good. However, admissions is very difficult in general and even tougher for international students. Admissions is also difficult to predict. However, a typical international applicant to a university on the “Harvard, Princeton, MIT, Stanford, Caltech” level is quite close to be the #1 top student in their high school (either overall, or in math and science particularly for MIT and Caltech) and somewhere around about 99% of international students are nonetheless rejected.
In spite of all of this, a few decades ago I did come to the US as an international student and get degrees from MIT and Stanford. My approach to applying to these schools can be summarized as “do what is right for you, and do it very well”. This is my understanding of the approach also recommended in the “applying sideways” blog on the MIT admissions web site. It is my understanding that this approach also applies to other top schools. Two close family members did use this same approach to get accepted to very strong graduate programs in the US, but (other than our grades) what they did that was right for them was completely different from what I did that was right for me.