Historically, checks were made of paper and required the user to physically cut the check from a checkbook. This process involved cutting the check along a perforated line, separating it from the rest of the checkbook.
Therefore, "cut a check" literally means to physically detach a check from a checkbook and is a metaphor for the act of paying someone.
While checks are becoming less common due to the rise of electronic payments, the phrase "cut a check" remains in use as a colloquialism for paying someone or making a payment.