[Question] How are transactions verified? (detailed answer please)

Please do not answer this with something high-level like 'Proof of Work'.

From what I understand, transactions are verified and added as a block to the blockchain once someone wins a proof-of-work. Essentially, they guessed a number faster than everyone else so they get to add the block the the blockchain. My question is, how are they verifying that the transactions within the block make sense (i.e. how are they checking that the people who are sending the bitcoin actually have the bitcoin in their wallet). I have some friends that do mining and they certainly have never considered this, they just use their GPUs to help others guess numbers in hopes that they win some btc lol