Firstly, I find the hostility against many worlds interpretation inadequate. As far as understanding quantum mechanics goes, the case is far from closed. I believe that it is unlikely, that this question will be answered near future (and it is plausible that it will be never resolved). Nevertheless, something being inherently hard should not suppress our thinking.
Secondly, why do I believe that many worlds interpretation is a sound approach? Before even hearing the MWI-word, we were talking about quantum entanglement in our department coffee room's blackboard (some 7 years ago) and I realized that measurement can be explained by quantum entanglement of the observer and the system. I think most physicists are too busy thinking about real problems (as they should), that they never sit down to think about these fundamental issues. (Also, one of the reasons is probably the choice of name and the stupid splitting movie film picture in Wikipedia).
Thirdly, I will have do define carefully what MWI is. As is correctly stated in the other answers, there are various definitions (the vicious say that there are as many definitions of MWI as there are supporters):
First thing you have to understand about MWI that it is formulated by a PhD student 60 years ago. The decoherence will not be invented in 15 years and the current main-stream interpretation of quantum mechanics is the Copenhagen interpretation with unintuitive wave function collapse postulate. Still, I find the thesis to be quite remarkable (Everett is also cited by decoherence paper, but I cannot access it so I do not know if it is in good or bad).
In Everett's thesis, he defines two ways of changing the wave function (as listed by von Neumann).
- Process: Suddenly, by assigning the system into eigenstates with probabilities $|<\phi_i|\Psi>|^2$.
- Process: Via unitary evolution, according to Schrodinger equation.
He lists several alternatives, but sticks with alternative 5:
To assume the universal validity of the quantum description, by the complete abandonment of Process 1. The general validity of pure wave mechanics, without any statistical assertions, is assumed for all physical systems, including observers and measuring apparata. Observation processes are to be described completely by the state function of the composite system which includes the observer and his object-system, and which at all times obeys the wave equation (Process 2).
As far as I see it, this is the crux of many-worlds interpretation. There is no need to consider a) unexplained sudden changes to wave function b) probabilistic rules of resetting quantum simulations at certain moments. Here a) and b) differ by whether the collapse is real in other interpretations.
Here is another quote:
We have seen that in almost all of these observer states it appears to the observer that the probabilistic aspects of the usual form of quantum theory are valid. We have thus seen how pure wave mechanics, without any initial probability assertions, can lead to these notions on a subjective level, as appearances to observers.
Here is one more:
We have shown that our theory based on pure wave mechanics, which takes as the basic description of physical systems the state function - supposed to be an objective description (i.e., in one-one, rather than statistical, correspondence to the behavior of the system) - can be put in satisfactory correspondence with experience. We saw that the probabilistic assertions of the usual interpretation of quantum mechanics can be deduced from this theory, in a manner analogous to the methods of classical statistical mechanics, as subjective appearances to observers - observers which were regarded simply as physical systems subject to the same type of description and laws as any other systems, and having no preferred position. The theory is therefore capable of supplying us with a complete conceptual model of the universe, consistent with the assumption that it contains more than one observer.
As far as I interpret this, Everett is saying that probabilistic interpretation of quantum mechanics is emergent from properties of unitary evolution of the wave function. This is appealing in many ways. First of all, wave function collapse as a phenomenon is explained. It is an entanglement between observer and the system. When added with decoherence, one has a theory of measurement which is complete and contains no awkward collapse postulates.
Finally and foremost, and this is why all the fuzz, here comes the many worlds part. Pure wave mechanics, also for observers, means that there are finite amplitudes in many different observer states simultaneously. Thus, if with a world we mean all things we can get information about, we will see, that the scientist who measured spin up will never communicate with the scientist who measured spin down. With orthodox interpretation, both of these scientists still exists in the world wave function.
Now, finally, and unfortunately at so late part of this answer because of all the fuzz, we can come to your questions:
MWI states that in some "worlds" the particle goes through one slit, and in others it goes through the other. If this is so, why do we get an interference pattern?
It does not state that at all! Exactly the opposite. It states all the regular things about two-slit experiment which can also be stated with probabilistic interpretations: Only, if one measured from which slit the particle goes, one does not get the interference pattern. Probabilistic interpretations call this collapse. MWI says that there is a quantum entangled state $|a>|A>+|b>|B>$ where small letters are slits and capitals observers reading a or b from their measuring apparatus.
Is MWI deterministic?
Yes, everything will be governed by the unitary evolution of the wave function. The Schöringer equation is of first order in time and thus exactly solvable when given a boundary condition (say wave function at surface t=0). This means that the system is fully deterministic. To elaborate further, in many worlds interpretation the unitary evolution of the wave function produces collapse of wave function only as an emergent process (thus abadoning process 1, as listed above).
To conclude, MWI should be treated as a historical way to modern understanding of quantum mechanics and measurement. In it's 60's formulation it is outdated, but the concepts still hold. I find it funny, that the proponents of probabilistic theory seem to find the MWI-theory non-sense, even their own interpretation can be derived MWI. It remains to be seen if future experiments can shed light into these interpretations. For example, although very unlikely, it would be very exciting if the progress in quantum computing would hit unexplained mysterious limitations requiring new theories. As far as the dislike towards this theory goes, it is probably related to fact that discussing interpretations is mostly hobby to everyone. There are very few who actually does research on this fields, and could comment on the latest events. The rest are probably like me: when I go to work tomorrow, I will wonder about theoretical modelling of plasmonics in photovoltaics, since that is what I get paid to do.