The theory of multiverse in short means that besides our own universe there are multiple universes that are similar but different from ours. But how and why do we think this is the case? Short answer is math predicts it like it predicted the theory or relativity and the idea of the expending universe, which were both later confirmed experimentally.
How to visualize the multiverse? The only thing we can do is compare it to some earthly objects like a foam bath or a boiling pot of water. Each bubble appears out of seemingly nowhere thanks to quantum fluctuations. Some of the bubbles collapse immediately, others expand infinitely like ours.
Watch a video with Brian Greene, American theoretical physicist and string theorist, explaining the concept of multiverse.