They don't check every public submission. From what I've seen they just produce a fact check article every now and then on something a politician has said or if there's a debate for example.
Fact checking is far less time consuming than writing an actual news article. There is no "finding the story" (which is easily the most difficult part of journalism), getting quotes from sources, ensuring balance, garnering a photograph if necessary, etc.
Fact checking is a simple as finding out if a statement made by a politician is correct. For a seasoned journo this should be a fairly trivial task tbh. Depending on what the statement is pertaining to, you just have to ask yourself where you'd find the truth, and it invariably would be in one of the following:
If the info is already out there, it can literally be as easy as cracking open the right report and hitting ctrl + f.