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Well, my degree is in astrophysics, so I'll give you a few comments here;

For an explosion to happen, one or more forces have to act upon another group of forces.

The Big Bang was not an explosion, instead it marked the beginning of the expansion of the universe. This may be a subtle, though important distinction. To explode there must be something to explode into.

So, as the universe has expanded, things get farther from each other, and to a casual observer from any spot in the universe, on the largest scale it would appear as if there was an explosion centered on where they are observing from.

Think of the universe as a piece of raisin bread. You put it in the oven and the dough starts to expand. The distance between each raisin increases because the dough between them has expanded. In this analogy, the dough is the fabric of space and the raisins are galaxies (well actually groups of galaxies).

In all seriousness, there has been a movement to rename the Big Bang theory because it causes just this confusion.

if all matter wasn't incinerated from the start

Well, it was actually, or at least pretty darn close. The 'matter' extent pre-Big Bang we have no idea about. The stuff shortly after was extremely exotic, and can only be recreated in particle accelerators now, if at all.

As time went on most of the particles settled into matter and anti-matter hydrogen atoms which obliterated each other until only the tiniest fraction of atoms remained, with matter oh so slightly more common then anti-matter and hence virtually none of the latter still exists.

While the Big Bang theory isn't perfect, it does fit what we have observed fairly well. It is also accepted by most everyone in the field by they conservative or liberal, theist or atheist. Big Bang theory is not some politically correct concoction.

If you really are interested, I'd suggest you ask your freshman science teacher, or your local librarian, for more you can read about on your own.