I am trying to accept a JSON payload that is supposed to create new assets in the Asset sObject. I'm trying to wrap my head around how to expose the API to the server that is sending the POST request. Are there any examples on how to do so and how to accept the JSON payload coming in?
The Salesforce API Reference has specific examples of every type of API call you can make, including, for example, the Asset resource. If you can control the JSON, this would be the ideal solution, as it involves writing no Apex code at all. If you need explicit parsing, you can look at the Apex REST documentation. It has several different modes of operation, including passing in method parameters that can be automatically parsed, or manual parsing with JSON support methods. Basically, there's several different choices to choose from, depending on what your JSON looks like, if you can configure it how you like, and the eventual design goal. You'll need to do some additional research to figure out which approach will be best for your situation.