Imagine this: You are BigCorp USA Inc. and have a bunch of internal SOAP-based web services that are 10 years old and cost a fortune to maintain.
You hear someone will pay you to use them, but SOAP sucks. So you could either go and change your APIs (and probably break a bunch of internal processes) OR you could use a gateway to sit in front of them and translate in bound / outbound requests. Now you can charge people for your APIs!
Scenario 2: If you're an indie dev/ startup but dont want to run EC2 infrastructure, Lambda + Gateway = Cloud service, without any heavy lifting, servers and cost effective. It's development lego. Kind of like Parse.
Scenario 3: You have an API for your app, now you wnat third party devs to use it.Your API is not set up for that kind of access (multi-user security, throttling, quotas, analytics etc.) - solution: shove a gateway in front of it.