What starting point are you looking for? In other words how much background do you need? I don't have a degree but I can try to give you the best overview I can find.
I've heard a lot of these theories are bogus, and if someone knowledgeable enough about the math can comment on that. Is it bogus armchair physics or is there real math supporting the idea that there are multiple universes many much like ours?
Despite using quantum mechanics everyday as part of my research, I'm by no means an expert in fundamental issues surrounding the interpretations of quantum mechanics, such as the multiverse. These arguments and interpretations (I would agree with Mindwolf that these are not theories in the usual sense, more interpretations of the existing theories) are more philosophical in nature. The Many-Worlds interpretation dates back to Hugh Everett III (who, in an unrelated note is the father of the lead singer of the Eels) and his PhD thesis. I would say these things are still very much on the fringe of physics and not the conventional or common interpretation of quantum mechanics. I don't think the people mentioned in the article are bogus in any sense, and certainly not armchair physicists!
Bogus isn't a proper word to describe it. The fact is that they do not count as theories. They are hypothetical, meaning that they are a proposed answer to a perceived problem. Now granted there is a lot of fierce debate over them. And a lot of physicists have called them bogus, but not from a scientific standpoint. Have you checked out the Wikipedia entry on it? It has some good information though I don't know how TL;DR it is.
So they are hypothetical answers to problems being suggested by the math? What is the confidence level that we're understanding the math and that these theories are even likely?
Well they all fit the math. The problem is there is not enough that we know to completely flesh out the hypothesis. When you add to the fact that these concepts are not even testable yet adds further chaos. Testing is what verifies the math. When you can't test, you don't really have anything.
Why can't we test? What is stopping us from figuring out how to interact with parallel universe? If it isn't possible, why?
The simplest answer is, we don't know how. Don't forget, we just proved the Higgs Boson. And we had to build a GIGANTIC particle collider to do it. And what did we find out? More questions. The Standard Model calculated that the Higgs should be at energy level A. String Theory says level C. The LHC found it a level B. Right in between the two theories. Saying this is an ongoing process would be an understatement. It's also what makes it the most exciting IMHO.
Just to take you up on your comment about the Higgs bosons, what you say is not really correct. Firstly, as far as I am aware, the mass of the Higgs boson is completely compatible with predictions from the standard model. Secondly, I don't think string theory has any predictions to make about physics at energy scales being tested by the LHC, and certainly didn't have any concrete predictions for the mass of the Higgs (after all, one of the main problems of string theory is that there are many string theories and we have no idea which is the correct one, if indeed one is correct). Thirdly saying The Standard Model calculated that the Higgs should be at energy level A
is not really true. In the Standard Model, the various parameters (such as "bare" masses of the particles and "bare" interaction strengths) are input parameters which one plugs in by hand; there is no calculation directly of the masses (although one can compute corrections to the "bare" masses due to interaction in terms of the input parameters). What one can do is vary, say, the mass of the Higgs and see if one gets computed quantities compatible with the observation; it doesn't directly "calculate" the mass (sorry if this is a bit pedantic).
That's interesting. Do you have any links to youtube videos on the topic?
Check out a documentary called Particle Fever it chronicles the discovery of the Higgs.