When you go fishing, you should bring the licence number you get when you buy the licence along with your personal ID. In fact, it is enough to memorise the number of the licence, but most people bring the print out, when they buy online, or the receipt from the post office, etc.
If you fish without a fishing licence, and you are inspected, you will have to pay a fine of 740 DKR (DKK 1,200 for hobby fishing with passive tackle).