Well, first of all, "branching" into different "universes" is a simplification, but I'll put that aside as that's a rather difficult area to explain. A gross simplification is that whenever an experiment can have two different results, each result exists, and the degree to which each exists is proportional to their probability.
So, let's say you have an experiment in which the Copenhagen interpretation says that you have a 70% chance of measuring a particle as spin up, and 30% chance of measuring spin down.
What MWI says is that the initial state can be decomposed into a quantum state in which the particle is spin up, and a quantum state in which the particle is spin down. As the states evolve, they interact with the rest of the universe, and so the states encompass the entire universe, including the experimenter. The first state ends up describing a universe in which the experimenter observes a spin up participle, and the second one a universe where the experimenter observes a spin down particle.
MWI also says that the number 0.7 is associated to the first state, and to the second, 0.3. These numbers are sometimes referred to as "measures", "weights", or "probabilities". Now, where things get a bit fuzzy is what these numbers "actually" are. The simple answer is that they are "probabilities"; the first number says that the "probability" of finding yourself in the first state in 70%. 28
But that just raises the question of what "probability" means. There's no physical "thing" that the word "probability" refers to. One can say that it refers to long term trends; if we had 1000 experiments with "essentially the same" setup (whatever that means), then we should expect to find ourselves observing spin up around 700. More precisely, observing spin up 700 times is the most likely result. But that just defines it in terms of probability again.
Classical physics says "If you do $X$, then $Y$ will happen". Copenhagen interpretation says "If you do $X$, then $Y_1$ will happen with probability $p_1$, and $Y_2$ will happen with probability $p_2$". MWI says "If you $X$, then there's this quantity 'measure' that has a value of $p_1$ for $Y_1$ and $p_2$ for $Y_2$". 28