Since a standard deck has 52 cards, and you need 5 cards to complete the hand, the probability of getting one of the cards you need first is 5/52. You then need to get the second, third, fourth, and fifth card from the deck and they would have probabilities of 4/51, 3/50, 2/49, and 1/48 respectively. This is because you remove one of the cards you need as well as a card from the deck overall. Since it is a series of events, you multiply all those probabilities together to get the total probability of that being the overall outcome.
This is the probability of a royal flush in any single suit. There are 4 suits overall, so multiply by 4 to get your total probability.