Okay, so imagine you have a phone, and you want it to do something when someone calls you - like play a special ringtone or show a picture of the person calling. But how does your phone know what to do? Well, that's where the telephony application programming interface (or API for short) comes in!
An API is like a set of instructions that tells your phone how to work with something else. In this case, the telephony API tells your phone how to work with the phone network - the system that lets you make and receive calls.
So when someone calls you, the telephony API tells your phone to ring and show you who's calling. But it can do a lot more than that too! For example, some apps might use the telephony API to let you record calls, or send text messages automatically when you miss a call.
Basically, the telephony API is like the boss of the phone network, and it makes sure that your phone does what you want it to do when you're using it to call or text people.