It's a fact; overpopulation is a problem here on Earth. While it's not necessarily as front-and-center as it was back in the 70s, it remains that the sheer number of humans will be the ultimate cause of environmental issues to a greater and greater degree. So perhaps it's not much of a supposition to suppose that there are a lot of people who, given the opportunity, would pack up and move to an offworld colony. Even more if said colony is on another habitable planet, rather than some L5 space station or Martian dome. Granted, this wouldn't realistically make a dent in population; not unless you've got a fleet of ships capable of lifting millions of people per day, and that in itself causes its own problems - it's hard to imagine a society that could absorb millions of newcomers just like that.
That doesn't happen too much in science fiction, either - colonies tend to follow the pattern of the European colonization of North America, with massacre and abuse of the natives optional depending on what point you're trying to get across with the story. When colonies do happen, though, they tend to happen fast. It's hardly unusual to encounter a setting where humans have been exploring the cosmos in earnest for a short time, and yet near space is peppered with colonies: Mass Effect is just the most recent prominent example of this, and possibly one of its purest distillations.
How realistic is this, though? Or, rather, how realistic is it that this sort of policy would be successful, and not fraught with high-profile incidents that would put the brakes on reckless settlement? For the purposes of this argument, I'll overlook the unlikelihood of there being multiple Earth-compatible planetary environments just there for the choosing. Really, how safe would it be if the immediate response to the discovery of a habitable planet was to settle it without delay?
Keep in mind that the challenges involved in settling a new world are ones that humans have not faced in more than ten thousand years, when people first crossed the Bering land bridge to the American continents - though I suppose you could also argue the Polynesian settlement of New Zealand eight hundred years ago had the same aspects as well. Any planet without a preexisting sapient population would be wilder than any part of Earth has been for generations, wholly untamed.
Plus, there are the dangers that would come from it being, you know, an entirely different planet, and the regular rules of Earth no longer applying. At least the Polynesians or the first inhabitants of North America would have had some idea of what to expect. A new planet could have any number of hidden dangers, from extreme seasons to hostile life forms to environmental poisons that aren't immediately obvious. A more appropriate trajectory, I think, would be to precede any planetary settlement with a long period of observation and initial exploration; enough so that if the planet does try to kill you, it will at least be in a way that doesn't happen every year or every other month.
Remember, it took more than a hundred years for European presence in North America to really stick - and don't forget about places like the Roanoke colony, which just up and disappeared. New planets would represent incredible investments - but one bad one early enough could wreck the whole thing.