If you treat the universe as an isolated system, then an outside observer will assign it a universal wavefunction which evolves under the Schrodinger equation and never collapses. The problem is that we -- by which mean the people who run experiments on quantum systems in the lab -- are not outside observers to the universe; we are definitely inside the universe. The universal wavefunction in principle contains the information about the quantum state of all the particles, including those which make up ourselves. But somehow you have to relate this to our subjective experience, and in particular, to the measurement outcomes that we observe when we run certain experiments in the lab. The Everett picture is one way to do this. I sometimes hear people say that Many-Worlds is a "logical consequence" of the universal wavefunction, but I strongly disagree; it is a proposal for how to assign physical meaning to the universal wavefunction.
Now, what is true mathematically (in any interpretation of quantum mechanics) is that if you start with a superposition of two macroscopically distinct quantum states, for example
$$\frac{1}{\sqrt{2}} ( | \mbox{atom decayed} \rangle + | \mbox{atom not decayed} \rangle)$$
and allow the system to interact with its environment, then the combined wavefunction of the system and environment (which is a part of the universal wavefunction) will decohere. What this means is that by the time the influence of the superposition has reached macroscopic scales, for example
$$ \frac{1}{\sqrt{2}} ( | \mbox{cat dead} \rangle + | \mbox{cat alive} \rangle)$$
or even
$$ \frac{1}{\sqrt{2}} ( | \mbox{the experimenter has opened the box and seen a dead cat} \rangle + | \mbox{the experimenter has opened the box and seen a live cat} \rangle)$$
there is no feasible operation that will ever distinguish this state from a classical statistical ensemble of cat dead with probability 1/2 and cat alive with probability 1/2. Thus, "cat dead and "cat alive" effectively forms two completely non-interacting "branches" of the universal wavefunction. So far, this is simply a mathematical statement about the structure of the wavefunction.
What the Many-Worlds interpretation does is it assigns physical meaning to these branches within the universal wavefunction -- namely, each branch describes a physical universe which actually exists, and the two universes both split off from a single universe. In each universe, the experimenter sees the corresponding result (dead cat or live cat). Now this is really a huge interpretative leap. But it might well be the only way to extract anything meaningful out of the universal wavefunction.
