In easy words, an API gateway solves a particular problem. The problem is that an API provides a bunch of "microservices" (little pieces of software that each do one specific thing) which have to be accessed individually. Each one of those services requires a "round trip" from the client (say, a web browser) to the server. This round trip takes up a lot of overhead (processing time).
So, it reduces overhead to have a piece of software at the server end that can take multiple requests for microservices from a client, access each of them on the server side, and package up all the results of that access and send them to the client in one trip. That piece of software is an API gateway.
This looks like a good basic explanation to me. Perhaps you can read it and then ask more specific questions about parts that you don't understand, if you have any.