The Many-Worlds Interpretation is a realist interpretation of the universe, but not a realist interpretation of the world.

The Many Worlds Interpretation (MWI) is a deterministic, realist interpretation of quantum mechanics (QM). MWI starts with two postulates

1. The universe is described by a quantum state, which is an element of a kind of vector space known as Hilbert space.
   
2. The quantum state evolves through time in accordance with the Schrödinger equation, with some particular Hamiltonian.
   

From the point of view of MWI, the quantum state of the universe (also known as the Universal Wave Function) is the thing that is "real". It evolves in a locally deterministic way.

What happened to the "other worlds"? Why is it even called the "many worlds" interpretation? Other worlds are not postulated by MWI, rather they arise naturally from an understanding of the behavior of the system based on just the two things we do postulate. And what about "decoherence"?

In MWI, decoherence is said to occur when the phase angle between components of the quantum state are sufficiently orthogonal that, for practical purposes, they do not exhibit interference. The fact that this occurs is a consequence of the underlying math. This happens naturally when information about quantum interactions (e.g. the result of a quantum experiment) spreads into the environment through further interactions (e.g. because the result is displayed on the measurement instrument, and photons from the instrument's display reach the experimenter's eyes, the walls, etc). Once that occurs, we can analyze the orthogonal components of the quantum state in isolation. These orthogonal components can be interpreted as independent worlds, or alternative futures of the world, each representing the future following one possible outcome of the interaction (e.g. measured result).

In practice the phase angles are never completely orthogonal, because the spread of information into the environment is limited by the speed of light; there are sufficiently distant regions of the universe where the components may interact. So the meaning of decoherence is interpretational: it depends on what we mean by "sufficiently orthogonal" and what the "practical purposes" are. If we are only interested in what happens in our experimental laboratory, the behavior of distant reaches of the universe in the distant future can be treated as irrelevant.

This is no different from saying that, for sufficiently small velocities, mechanical systems obey classical rather than relativistic behavior. What is "sufficiently small"? It depends on the context. Nature does not care what we mean by sufficiently small, it always obeys the relativistic rules. But the concept of classical behavior allows us to simplify our calculations (at the expense of introducing a small inaccuracy) to improve our understanding of the system.

So it is for decoherence. It is not a term that is rigidly defined in the theory, but (like "non-relativistic velocity") is rather a concept for a simplifying assumption that we use to understand the behavior of the quantum state. Decoherence cannot properly be said to occur at some particular time, like quantum collapse in the Copenhagen interpretation. It is not an "event that happens", but rather a change in the way we interpret the meaning of the quantum state from one time to another.

One accepts MWI at the expense of rejecting objective reality as we know it. When we open the box to see whether Schrödinger's cat is alive or dead, we become entangled with the cat's quantum state. If we see that the cat is alive (as we hope), we cannot say that the cat's status of being alive is a fundamentally true fact about the universe. Rather in the quantum state of the universe, there are nearly orthogonal components that can be interpreted as two versions of our world, one in which we observe the cat being dead, and one in which we observe a living cat. One of them feels more real, somehow, but each component describes a version of us who thinks it is he who is observing the true world. MWI doesn't designate one of these components as somehow more real than the other, and thus we can think of them as separate worlds, or futures.

[MWI] predicts that we will think and claim, that we do not observe superpositions at all, even when our own states are highly indefinite, and that we are simply mistaken in the belief that we see a particular outcome or other. That is, it preserves unitary [deterministic] QM – at the expense of a skepticism that "makes Descartes’s demon and other brain-in-the-vat stories look like wildly optimistic appraisals of our epistemic situation" [The Ashgate Companion to Contemporary Philosophy of Physics page 43]

This is like Einstein's principle of relativity in another way, too. In MWI, the meaning of the world is relative to the observer. If you ask whether the cat is alive or dead as a property of the universe, the simple answer is that the cat is in an indefinite state. To give a more definite answer we would need to know which (mostly) orthogonal component of the quantum state you're asking about. Which world did you mean?