It would be more more fair to say that a president "doesn't not have" the right, rather than "absolutely does".
It's a grey area, and would probably decided in a court if it was attempted and challenged. The Constitution doesn't give the president explicit power to do so, but also doesn't explicitly say a president can't do it.
Pardon Definition: Article II, Section 2 of the Constituion defines the power of the Presidential pardon:
...and he shall have Power to grant Reprieves and Pardons for Offenses against the United States, except in Cases of Impeachment.
Not much there, and since it just says "grant Reprieves and Pardons for Offenses" and not something like "grant Reprieves and Pardons to others for Offenses", it can be assumed he can pardon himself for any offenses against the United States.
Do note, that the pardon explicitly does not cover impeachment. It can be assumed that even if a president could pardon themselves, they wouldn't stay president for long.
Court Interpretation: Until such a pardon happens and is challenged in court, we won't know exactly if a self-pardon would be legal. However, this isn't the first time (and probably not last, unfortunately) that the idea of a self-pardon has come up.
Prior to resigning in the wake of the Watergate scandal, Richard Nixon, or at least his lawyers, examined whether a president could pardon himself. An Attorney General legal counsel, Mary Lawton, came to the conclusion that it is not possible to self-pardon:
Under the fundamental rule that no one may be a judge in his own case, the President cannot pardon himself.
The necessity doctrine would not appear applicable here. That doctrine deals
with the situation in which the sole or all judges or officials who have jurisdiction to decide a case are disqualified because they belong to a class of persons who have some interest in the outcome of the litigation, thus depriving the citizen of a forum to have his case decided.
Essentially, Lawton provides two reasons as to why a president cannot pardon themselves: because nobody should be their own judge, and because a person cannot receive a fair trial if the judge has some interest in the outcome. However, this is just her interpretation of the legalese, and a court may decide differently.
Regardless, if a president did pardon themselves, I'm sure the impeachment process would go a lot quicker than the court case that's trying to decide whether or not they were allowed to pardon themselves.