Ok, so the "check" terminology is clearly confusing. A "Check" is a resource you can earn. You get a free check at the start of a "session" (so you all have 1), and checks can be spent for a variety of things (there's a section on them). Improving a skill can be done by spending a Check and setting the XP to zero.
Besides the free one, how else do you earn Checks? Primarily through your weakness. Bring your weakness up, so introduce it into the story, and this will earn you a check, it may earn you more than one depending on how long the weakness stays around (the GM will never introduce your weakness, you always bring it on yourself). While exposed to a weakness your skills are all treated as rating 1.
Note: Checks and Twists section on pg 11 has the details