The Second Law of Thermodynamics states that the total entropy in a closed system must increase. The Earth by itself isn't a closed system; the Sun provides it with energy by increasing its own entropy by huge amounts via fusion reactions. That means that entropy in another part of the system can decrease without breaking the Second Law.
I've explained it more thoroughly here, but just think about it for a second. If order and complexity could never increase anywhere, you couldn't build anything, arrange anything or form a coherent thought. There must be a way.
It's unlikely that what exploded in the Big Bang was a total void. It was the entire current universe squashed into one tiny dot. We don't know how it got there; maybe it came from another universe, maybe something compressed the universe into a dot, or maybe it really did pop out of "nothing". That last one isn't quite as silly as it sounds, because some quantum theories actually do allow for it by positing that "nothing" is really a sort of quantum foam of potentiality.
We don't really know that the Big Bang was what created everything; we're just mostly sure that everything there is was in the Big Bang. It might have all existed forever, and just spent a bit of time squashed into the dot.
Thinking of the Big Bang as an explosion is a bit too simple. An explosion destroys things around it; since everything was in the Bang itself, there was nothing around it to destroy. Once all the debris was floating free, gravity brought some of it back together to form rocks and stars. Stars create vast amounts of entropy, so any rock receiving energy from a star is part of its closed system and a certain amount of order and complexity is free to emerge there. This is how microbes were able to come about (though it's not exactly how they came about; that's a bit more complicated).
Plants actually evolved from seagoing creatures. Once the oceans were teeming with ultra-primitive life, masses of it was bound to wash up on the shores of Pangaea (the single pre-drift continent). Most of that organic matter would have died, but a tiny fraction of organisms would have been able to use their existing abilities to sink safely into the sand or soil, and stick one piece up to get some sunlight. It was natural selection in all its glory; if you throw enough different kinds of crap at the wall, something is bound to stick.
Go ahead and reply with your objections if something still doesn't seem feasible. Don't worry about ticking us off. That's what we're here for.