ELI5 answer:
API is how programs communicate together. API is in computers world something like “we set up an an office on XY street, here are forms you have to fill if you want something and here is what we will send you back. And include this secret code, so that we know it’s you”
API subscription is that one company let other company applications to communicate with each other (sometimes for free, sometimes in exchange for some money).
In the case you are probably referring to, Twitter asked large sum of cash to allow Apollo application to talk to Twitter programs running on their servers
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