Usually, when you talk about observing a fast it is because the fast has a deeper meaning to it. I think we use "observe" instead of "carry out" because it shows that there's something behind the actions. We use "carry out" for something more task-related. Fasting, of course, has an element of doing something, but it is usually connected to a deeper spiritual (or at least sentimental) reason. I don't think that people would use "observe a fast" when someone is fasting for health reasons. For example, it is very popular in America to do a juice fast to detoxify the body. But we don't say we are "observing" this kind of fast. We just say we are fasting or detoxing.

I will guess the answer, someone can correct me. It's because you're NOT doing something. For example, if there was a rule saying "do not steal", and I "observe" that rule, you wouldn't say that I'm carrying out the not-stealing. If not, then the etymology is probably somewhere between "obligation" and "serve".

'Observe' in this sense means follow, or do as is required by law. We observe rules, we observe laws. It can also mean 'respect' in this sense. We don't use the word 'carry out' for religious duties, we say observe. 'Carry out' is for a job, a task, instructions in a work situation, for example.