If something presented as a fact is checkable, someone can confirm that it’s really true or correct by doing some research. Those dates and events you used in your history essay are checkable, so hopefully you didn’t make them up!
To check something is to verify it to make sure it’s true or correct. So checkable can mean "able to be verified," like the college degrees and work experiences listed on someone’s resume. If the details in a report are not checkable, you may have to take them on trust! But a check is also the little paper form you can fill out to pay someone with money from your bank account. So a checkable account is one you can write checks on.