Black holes do not die per se, but they are theoretically predicted to eventually slowly evaporate over extremely long time scales.
Black holes grow by the accretion of matter nearby that is pulled in by their immense gravity. Hawking predicted that black holes could also radiate away energy and shrink very slowly. Quantum theory suggests that there exist virtual particles popping in and out of existence all the time. When this happens, a particle and its companion anti-particle appear. However, they can also recombine and disappear again. When this process occurs near the event horizon of a black hole, strange things can happen. Instead of the particle antiparticle pair existing for a moment and then annihilating each other, one of them can get by gravity and fall into the black hole, while the other particle can fly off into space. Over very long timescales, we are speaking about timescales that are much much longer than the age of our universe, the theory states that this trickle of escaping particles will cause the black hole to slowly evaporate.