Every particle in the universe attracts every other particle, by gravity. There are other forces between them too, but over all, there's lots of mutual attraction going on.
Now, suppose you wanted to rip all the matter apart, so that all the particles were at infinite distances from one another, and there was no more attraction. You would have to add lots of energy, to move the particles away from each other.
It makes sense to think of the gravitational field between two particles as having negative energy, because you have to add energy to overcome it.
The idea of a "zero energy" universe is that the amount of energy you'd have to add to the universe to remove all the attractive forces in this way is equal to the amount of mass that you'd have, once all the particles are separate from each other.