It means to issue a cheque (or check) e.g 'To cut a check for $10,000' .It is American and not known in British English. Whilst some people think its origin is in the American military because the commander might 'cut an order' it seems likely that its origin is simply in the process of 'printing' a cheque by the use of a stencil, mimeograph machine or similar reproduction.This process involved the cutting of the lettering (as in 'to cut a stencil'). A company issuing lots of checks would have the blank check forms printed, using this primitive equipment, with the company's name as payer and perhaps other details ( payee's name, space for dollars etc ). So it would 'cut a check'. To 'cut an order' in the military, likewise involved, printing, running off copies for signature.
By analogy we say 'to cut a record' (music) because the machinery originaly involved cutting a groove, even though now we use purely electronic methods.