How will the topic of my Master's thesis affect my PhD?
In the US it should matter very little. Having a closely related area in mind might be an advantage, but changing might not be much of any disadvantage, within limits. In fact, having too narrow an interest might make it difficult to find an advisor in the US. Some flexibility is suggested.

Note that most beginning doctoral students in the US don't come with an MS or with much research experience. The first task is to pass qualifying examinations and some coursework is provided for that. Along the way you choose an advisor and a topic. But the topic has to be mutually agreed on. 

It isn't quite correct to say that all they care about is your ability to do research. But it is fairly correct provided that the fields you previously studied and the new one is fairly close. But expect to have to do some background work at least. 

And, even having a wildly different MS topic might not hurt your chances for admission. It would just add to the time to completion as it would for someone entering with a BS instead of an MS. But a lot of things will be looked at in any application: courses taken, grades, test results, letters of recommendation...