‘A’ has got 1BTC legitimately. He sends out a transaction to transfer that one bitcoin to ‘B’ with a 1BTC transaction fee added on. (i.e. a bad request.)
‘C’ is a miner who cares nothing for integrity of the bitcoin system and only cares about winning blocks and selling the BTC he gets as soon as he can.
What incentive does C have to pass on the free 1BTC and not include A’s transaction when putting the next block together?
Any block that contains an invalid transaction will be considered invalid as a whole, and every other peer/miner will reject it. A miner must check that every transaction included in a block is valid, otherwise there is a high risk that all the hard work required to solve the block is just being wasted.
Any miner who disobeys the validity requirements (by considering some invalid block as valid, or some valid block as invalid) will be completely disregarded by the rest of the network, defeating the point of any mining activity.
Note: Sending 1BTC with a fee of 1BTC is a perfectly valid transaction (even though it might be a mistake by A). Of course, doing that, A must own 2BTC and give them away, or the transaction can’t possibly be valid.