The answer is not really defined in 5e.
Before we talk about where Oathbreakers get their power from, we should discuss Paladins as a whole.
The Player's Handbook doesn't clearly define the source of a Paladin's powers, whatever oath they are attached to. Oaths might be "sworn before a god's altar and the witness of a priest, in a sacred glade before nature spirits and fey beings, or in a moment of desperation and grief with the dead as the only witness", which implies a diversity of power sources (divine, nature spirits/old gods, ancestral spirits, maybe just your own gritty determination). It's not clear cut because it's really up to the player to define where their power comes from. It's a space where the player can be creative.
We sometimes say their power comes "from the Oath itself" but that doesn't mean you have power just because you really want to, or generated by your pure belief -- rather it means whatever you've sworn the oath to, that's the source of your powers.
So what about Oathbreakers? Well, it kind of tells us right there in the DMG: Oathbreakers "pursue some dark ambition or serve an evil power." It's as much up to the DM or player to determine what power they now serve as it is in the more general case of Paladins as a whole. Most likely we're talking about dark gods, devils, or some such thing, but it could be the same natural or ancestral power they had before, twisted to darker manifestations.
Earlier editions are different.
In prior editions, Paladins were specifically empowered by the same kind of divine connection that Clerics have, which includes, in some editions, being allowed to revere a broad concept like Good or Law without declaring a specific deity. Paladins manifest that power in a different way, martial might rather than miraculous spellcasting, but it's explicitly the same source.
By default, paladins were what we'd now call a Devotion paladin -- the code of conduct was central to the class, though sometimes you'd have optional prestige classes to alter it. (I recall a particular prestige class that was the first version of what became the Oath of the Ancients.) A paladin who broke their oaths in previous editions would lose their powers until absolved in some way, which was often a point of friction between players and DMs who disagreed on what exactly the oath entailed and how strict it was.
In older editions, what we now call Oathbreakers would have been called Blackguards or (if you go way back) Anti-Paladins. They were generally reckoned to have sworn to an evil deity or a fiend of some sort, and had powers similar but opposite to a Paladin's. To the extent that 5th Edition's Oathbreakers are defined at all, it seems pretty much on the same line.