And if you go down that route, you pay for the increased deployment and maintenance complexity as well as the RTT to call the API service and consume the response as well as the CPU cost to render the page. There's no free lunch here. You can have both the API and rendering service on the same box, but now it's looking less and less clean and tidy, and more and more like a traditional web-app.
Your client (could be a web app consuming the api from the server) accesses the api like anyone else. You've created an api that others can build on - and you know that, because you're doing it yourself.
More to maintain. More cpu processing. Still a cleaner architecture though (compared to, say, having to maintain an external api for others to use while having a web app that talks directly to the db).