Beginners guide to connecting to external services via API

Beginners guide to connecting to external services via API

I am ok at writing functions, connecting to existing services and the likes from CRM.  The bit that completely escapes me is how to set up a new service.  I have read documentation and forum posts, but I just can't seem to get my head around it.  Does anyone have the patience to explain it to me in really simple terms?

Example service:  https://www.pdfmonkey.io/doc/api/generate-a-document/    
(although I'm hoping to be able to utilise these instructions for other services)

Any help is greatly appreciated!