I have some JS on my website which requests Data from my own PHP API and the API request data from external APIs (Google, YT etc.)
How can my own PHP API can verify, that the request is from MY Javascript and not from someone else?
If all you want to do is to simply prevent people from accessing your API through JS on other sites you can just set the CORS header Access-Control-Allow-Origin to your own domain.
header("Access-Control-Allow-Origin: your.domain.here");
Note that this doesn't mean people can't access your API through direct calls using software like Postman, python scripts etc.
If, for example, you want only yourself and/or specific users to be able to access the API you could look into php session management. This way you can create accounts which are allowed to access specific data on your API.
You could also use API keys to allow users to directly communicate with your API if needed.