A "Service" is when someone or something does something for you, usually in return for some fee.
A "Platform," in the context of IT and software (and especially the internet), is some IT infrastructure, generally a server and the software installed on it, that you can host something on, such as an app or a website.
A "Platform as a Service" is when someone else sets up your platform for you so that you don't have to do it yourself—you get access to the platform and can use it for your own stuff, but don't have to configure or maintain most of it.
