Black holes can "eat" Hawking radiation, but the radiation has to be directed towards the black hole before it can be "eaten". Black holes are pretty small, and space is very very very big.
So as a black hole evaporates, and gives off" Hawking radiation, some may eventually fall into another black hole, but most will not. It will just travel out into space, travelling forever (or until it is red-shifted to such low energy that it loses an individual identity - the so-called "heat death"). We can't picture this because our brains are not equipped to deal with the massive amounts of time. But on these time scales there is nothing very mysterious about a black hole evaporating and so vanishing. It is no more mysterious than a candle flame going out.
Here is a metaphor:
There are some children in a field, each child is carrying some balls (the children represent black holes, the balls are the mass of the black holes and the field is the universe).
The children are walking away from each other (the represents the expansion of the universe*)
Roughly every minute each child throws a ball in a random direction. (This is Hawking radiation)
If the ball passes another child, he catches it and adds it to his collection (Radiation falling into the black hole).
If a child runs out of balls, he sits down, he is out of the game and can't catch any more balls (this is the black hole running out of mass, and so evaporating).
Will the game end?
Most of the balls will never be caught by other children because they are not thrown in the right direction. Eventually, all the children will be sitting down and all the balls will be lost (This is the "heat death")
*This isn't quite right because it is an expansion, not walking from a centre, there is no centre to the universe.