Last year, famed physicist Stephen Hawking made headlines with this bold statement, "There are no black holes." Those words come directly from Hawking’s latest paper, but they are contained within a larger point involving the mechanics of a black hole and its famous "event horizon," the area thought to exist around a black hole from which nothing, not even light, can escape. To be clear, Hawking was not claiming that black holes don’t exist. Astronomers have been observing black holes for decades. What Hawking did was propose an explanation to one of the most puzzling problems in theoretical physics; How can black holes exist when they seem to break two fundamental laws of physics -- Einstein’s laws of relativity and quantum mechanics?
A black hole is an event in space-time where because of the presence of a large amount of matter, the force of gravity becomes as strong as it was back before the formation of the universe and the Big Bang, when gravity held all existing energy tightly in the form of a quantum singularity -- an infinitely dense and hot point of non-particle energy. The Big Bang marked the point where that energy was able to overcome the gravity for reasons yet unknown.
Inside the black hole, conditions are like that of the pre-big-bang epoch. Under these conditions, space-time cannot exist in the same way it does in the normal space-time we are used to. The physics won't allow it. We can model the physics by using laws discovered concerning both general relativity and quantum mechanics. We just follow the models backwards to mathematically predict what is happening inside a black hole.
As for infinities, black hole does not exist inside the universe (which is a closed system -- already proven) except in the abstract. An actual infinity would reach beyond the universe and therefore is not something that we could ever know as human beings. So don't waste your time with the "infinite universe." It's a fairy tale.
A black hole is extremely small in comparison to the singularity that gave birth to the universe. It was once theorized that a black hole could produced a white hole in another "reality," creating a new universe, but we figured out that a black hole, despite its tremendous gravity, is not strong enough to tear a hole in our universe and accomplish such a feat. So, no, black holes do not explode and create new universes.
It has been theorized by Hawking and others that black holes can create "baby universes." These are instances of particles which get ejected from black holes that have their own localized space-time different than ours. Not large enough to actually notice. They quickly evaporate and rejoin our space-time because there is not enough energy to sustain them.