What is the probability of getting two cards that belong to the same

The wording of the question is strange. I guess by 'belong to the same color', they mean that you either pick two red cards, or two black cards. In a standard deck of cards there are 26 red cards and 26 black cards -- note that you do *not* need to know anything at all about standard decks of cards on the GMAT.

Now the first card we pick doesn't matter. We only want to make sure our second card is the same colour as whatever we picked first. When we pick our second card we are selecting from the 51 remaining cards, and there are 25 cards left which are the same colour as the first card we picked, so the answer must be 25/51.

You've cited the OA as 'C', which is not correct. Answer choice C is the probability of picking, specifically, two red cards (or equivalently, it's the probability of picking two black cards). The correct answer is exactly twice as large as answer choice C.