The most typical way to profit from this, is to have quite a bit of ETC, and then sell it on an exchange or buy something expensive. This transaction would then end up on the block chain, and eventually be considered secure / part of history.

Privately, you are building a chain where said transaction did _not_ occur. Because your hash rate is high enough, you are generating blocks at at least the pace of the public chain. You keep mining your chain (without your transaction) privately until the bank transfer from the exchange goes through, or you are sure whatever you bought is on its way. Then you broadcast your private chain as soon as it has at least one block more than the public chain and voila, you have your ETC back and the USD/EUR/crypto you sold it for (or the product you bought). You can have your cake and eat it too. This is because the rule of most cryptocurrencies is that the longest chain is the truth, and everyone mines on top of the longest chain. If a longer chain appears out of nowhere, all miners will jump on top of this one. There is no such thing as "finality" in most cryptocurrencies, where something becomes actually (read: much more) irreversible.

Of course, doing this only makes sense if what you gain outweighs the costs of performing such an attack.

Note that you typically _cannot_ steal coins from specific wallets, as you do not have the keys to those coins. You can however censor transactions that you may not want, but the above is a common way to benefit from a 51% attack.