This may grossly oversimplify the process but it goes something like this.
1) First you must accept “vacuum” fluctuations, for our use here we will take that to mean in a vacuum as random very fast events where a Photon moving forward in time and an anti-Photon moving backwards in time are created (or at least some kind of particle anti-particle pair). They meet each other very quickly and annihilate themselves into nothing.
2) We can also assume that from our view they take next to no time to occur, and the energy that “appears” in the form of two photons is gone before we can see it, thus to use it seems that vacuum was still a simple vacuum where nothing really happened. Plus, we can speculate that these unseen events happen more frequently near larger accumulations of mass.
3) When these things happen near but outside a black hole, sometimes one of the pair is pointed toward the Black hole and goes in while the other “escapes”.
4) The one that goes in annihilates with someone inside the hole, thus reducing the energy/mass of the black hole.
5) Note, if the one escaping is the “anti” one moving “backwards” in time, that is only true in the space of the fluctuation, reference Feynman diagrams to understand why when it comes into our “normal” time we see it as a normal Photon moving forward in time.
This would give the appearance of energy come out of a black hole as it gets smaller! Hawking also claims that this process runs faster and faster as the black hole does get smaller until at some critical size the remainder of the black hole “boils” off in a flash.
Frankly you have to look at such a set up as just some wild theory in a desperate attempt to explain how the mass in a black hole can be returned to our universe and not be trapped there forever.
The last time we saw something this desperate was when some nut a few decades ago wanted to resolve problems they were having in balancing mass/energy conservation in nuclear reactions by just creating a unknown, unseen, and undetectable new particle. That was “NUTS” but, if I remember right that imaginary particle led to a couple guys getting a Noble, and the word Neutrino be added to our dictionary. That doesn’t mean Hawking is right, but the idea is as hard to prove wrong as it is to prove right.