The Big Bang didn't create organic molecules. Mostly what it created was hydrogen, along with a lot of weird things that aren't strictly important right now.
All that hydrogen drifted into clumps and accumulated through gravity. These became stars. It's true that we haven't seen this happen, because it takes millions of years, but we can see throughout the universe around us stellar nurseries in various stages of formation, and the physics is pretty straightforward - when you get enough stuff in one place there's only really one thing it's going to do.
These stars fused hydrogen into other elements. We know this happens because we can study the elemental makeup of stars through the precise spectra of light they emit. Fusion of lighter elements has been performed on earth (mostly in the process of making things blow up, but also for other types of experiments) and we know the conditions inside of stars allow for it. It's also the most straightforward way that the sun could emit as much energy as we know it does for as long as we know it has without visibly dimming or shrinking.
When big stars reach a certain point, they explode. We call these supernovas (or supernovae if you like Latin). We know they exist because we've studied the aftermaths of many of them. We know that the conditions during the supernova result in the fusion of a lot of heavier elements than you would normally get in stars.
All this stuff doesn't just vanish, of course. It hangs out in space and gradually accretes, just like how stars were formed in the first place. Some of it becomes new stars; some of it becomes planets. We know that supernovas can generate the right elements to make rocky planets, and even in the right concentrations to produce planets like our own. There's some confusion over precisely how it happens, but that's on the scale of modeling how many planets and of what size you get out of a cloud; the basic supernova remnant -> planets science is well understood.
Now, on some of these planets, we have certain conditions. There tends to be a liquid medium, floating on the surface, in either a single ocean or multiple oceans depending on size. They have an atmosphere. These parts of the planet are formed from lighter, more reactive elements (primarily carbon, hydrogen, oxygen, and nitrogen, at least on Earth). These four elements, and trace amounts of others, form most of the complex molecules you're probably familiar with, things like hydrocarbons (oil, gasoline, propane, etc.), plastics, and most of the molecules that make up the body.
The thing with these four elements is that they're highly reactive. Carbon and hydrogen both burn readily, and nitrogen is a major component of a lot of types of explosives. And the early planet was a goldmine of energy input: lightning strikes, lava and magma, impacts from meteorites, all playing on a primordial sea full of these reactive elements. We don't know exactly what reaction created the first amino acids, but all the conditions needed were certainly there.
