Re: Why did the human line evolve after 'the meteor hit' and not before?

Why did the human line evolve after 'the meteor hit' and not before?

Did we evolve from something that already crawled out of the water before 
the meteor strike or did we evolve from something that crawled out after the 
meteor?

Mammals (including humans), reptiles, and birds are all Amniotes, meaning that their embryos are surrounded by specialized membranes (including the Amnion) which protect them during development. All amniotes are descendent from terrestrial amphibians of the early Carboniferous Age (about 350 mya - million years ago). During the later Carboniferous (about 300 mya) the amniotes broke into two branches: the sauropsids, which gave rise to modern reptiles and birds; and the synapsids, which gave rise to the modern mammals. By the Permian Age (about 270 mya), the synapsid line had evolved into the pelycosaurs ("fin-backed reptiles"), including Dimetrodon, which occupied the top of the Permian food chain. Synapsids continued their rule through the early Triassic Age (about 230 mya) as Therapsids, which were outcompeted by the Dinosaurs, but not before giving rise to the first mammals which lived during the Jurassic Age (about 200 mya). So, mammals existed concurrently with dinosaurs throughout the Mesozoic Era which ended with the Yucutan Meteorite.

If humans evolved after dinosaurs were wiped out, what new conditions 
favored human life?  I realize not being eaten by dinosaurs favors human 
life, but there must have been something that favored dinosaurs over humans 
the first time around and humans over dinosaurs the second, i.e. atmosphere 
composition or climate.

Actually, you've answered your own question! The major difference between the Mesozoic Era (ruled by dinosaurs) and the Cenozoic Era (ruled by mammals) was climate: the weather has been much cooler during the Cenozoic than when the dinosaurs lived. Actually, several things happened at the beginning of the Cenozoic (about 65 mya), all of which seem interrelated. First, as was just said, the weather got cooler to the point that the Earth started having "Ice Ages". Second, angiosperms (flowering plants) became the predominant form of terrestrial plant life, probably because they could adapt to the cooler, drier climate. Third, there were no more dinosaurs: some paleotologists blame the meteorite; some blame their inability to digest flowering plants; some blame the weather. Whatever the reason, the extinction of the dinosaurs opened vast resources with little competition in a miriad of niches that were quickly filled by hungry mammals and birds (the warm-blooded decendents of the dinosaurs). The lack of predation also opened up new food chains which were likewise filled by mammals. With the cooler climate, the angiosperms, birds, and mammals completely took over the Cenozoic Era.

Regarding human evolution, our pre-hominid ancestors separated from the Chimpanzee lineage about 5 mya - about one-tenth the time of modern mammalian evolution - with the earliest Australopithecine hominids appearing in the fossil record about 4 mya. Human evolution was influenced entirely by competition with other mammals, especially since the effects of the meteorite had dissipated some 60 million years earlier.

Is there DNA evidence supporting my question one way or the other?

By now, there is almost more DNA evidence than there is fossil evidence. Rather than try to explain it all here, I would send you over to the Sequence Comparison Tutorial at Cold Spring Harbor, which gives you a hands-on example of the supporting DNA evidence.