First, from a cryptography point of view, transactions have to be signed, and the other nodes in the network will be able to detect and reject transactions for which the signature was forged. Second, from an incentive point of view the miner will have invested some time in solving a block that will, as a result of the previous point, not earn him anything. So we can expect he won't try in the first place.
Another way to 'award themselves with any number of bitcoins' would be to create, rather than steal, the bitcoins. Indeed, mining is also about creating some bitcoins (currently ca. 25). But if a miner was to create more than 25, the block would again, not be acknowledged by the other nodes in the network.
I think the point is that a race is not won by "adding the next block" but rather by adding a block that will be considered by the other nodes to be valid.