From my understanding, a massless black hole continues its existence because it is actually a massless elementary particle.
The way this works is because in string theory there is something called a conifold transition. This transition is the tearing of a space within a Calabi-Yau space (the six-dimensional shape that the extra dimensions called for by the theory are wrapped up in) and the subsequent patching of that tear.
As space tears, it shrinks down to a single point, tears, then patches itself up, regrowing to the size it had been. The tearing of space would typically cause a major catastrophe in our universe, but it doesn't because the rest of the Calabi-Yau shape can be shielded from the space that experiences the tear, as Strominger discovered, by a three-brane that wraps itself around the tearing space. This three-dimensional brane-wrapped space has the gravitational field of an extremal black hole. So when the space pinches down to that single point, the brane also shrinks, and what we perceive as an extremal black hole becomes massless. This happens because as the space gets smaller, the brane has less volume to cover, so its mass lowers, which is also happens to be the mass of the black hole.
Once the space reaches the state of being a single point, it tears. After this happens, it begins to patch itself up, causing the addition of a new massless string vibration pattern because of the nature of the growing space that is filling the tear. Due to this patchwork the black hole is now perceived as a massless particle. Unfortunately I don't know why just one is added, but I believe it arises from the math used to describe the transition.
But what is important is that the extremal black hole, which becomes massless, transforms into a massless particle. So in essence, the black hole doesn't continue its existence as a black hole because the process that rendered it massless also turns it into a particle, which can easily exist without a mass.
This process was discovered by Strominger, Morrison, and Greene. For a very detailed explanation of conifold transitions, read Greene's book, the Elegant Universe. There's a whole chapter on the process. I hope this explanation helped!