The Big Bang happened quite a few years ago. At first, it only produced elementary particles. Shortly thereafter, some of those particles combined into protons, neutrons, and electrons. Around that point, there was quite a lot of hydrogen, but not much else.
Eventually, gravity caused the hydrogen to form clusters. These clusters became stars. Since then, stars have lived their entire lifetimes–forming, burning, then dying. Some stars die with a whimper–along the way, they generally produce medium-to-light elements such as iron and carbon.
Some stars die spectacularly, in the form of a supernova. The red giant stars that eventually supernova are the source of almost all of the heavier elements such as lead.
So, the Big Bang is really outside the scope of your question. Planetary formation begins inside stars.
Pretty much everything heavier than iron was formed in the heart of a red giant. Iron was formed inside lesser stars. When a red giant dies, it explodes violently–a supernova–and all the material that made up the giant, plus any nearby material formed inside of normal stars, gets blown all over that part of space.
But gravity is still in effect. These bits and pieces draw each other, eventually forming a new smaller star. That star becomes a central point of gravitational focus–every bit of matter is both drawn to the star and drawn to every other bit of matter. A disk of dirty gases forms, all of it revolving around the star. With time, clumps form in the disk.
By now, there are several effects in play: most of the matter is still somewhat affected by the momentum it had when it was ejected by the supernova, the new star is pulling everything toward itself, everything in the disk is pulling on everything else, each clump of matter is pulling all nearby dust and gas toward itself, and the whole mess is spinning arond the new star.
With time, each of those clumps grows larger and begins to rotate on its own axis… again, due to gravity and momentum. Everything in the above paragraph is still occurring (though there is less free dust and gas), but now we can also add in the spin of the individual bodies (planets, moons, whatever the clumps are growing into).
We know that gravity draws heavier elements toward the center of a body, while the lighter ones stay closer to the surface. This causes each planet or moon to have some internal “churn”… don’t forget that the object is also both rotating (in a small circle) and revolving (in a big circle). That’s what causes them to become spheres.
Note, though, that not all bodies in space are spheres. There is a certain mass required for a body to overcome its own tensile strength and take on a spheroidal shape. In fact, one requirement for a body to be classified as a “dwarf planet” or “planet” is that it has been rounded by its own gravity. Many objects haven’t.
You asked for a simple explanation, and this was about as simple as it gets. I haven’t mentioned hydrostatics or various tidal forces or even centrifugal effects, which also play a role in shaping the heavenly bodies. One thing of note, though, is that most stars aren’t spheres… this is due to all the reasons I didn’t mention.
Hope that helped some.
ETA: I see that I was beaten to the punch by the pros. 
