There was no need for Europeans to travel to the new world, cause they had enough over there.
I don't think the European colonization of the new world is a very good analogy here, for several reasons
1) There was a clear financial motive towards exploration; find a new trade route and reap incredible monetary gains. Bring back gold and spicies and be handsomely rewarded. I don't see the same rewards for going to a new solar system - the energy required to go, stop and then return are incredible. If you plan on bringing anything back (e.g. rare minerals) those will incur another huge energy penalty.
Colonization was similar - we knew where we were going was already immediately suitable for living, and not appreciably different from homesteading elsewhere in Europe.
2) The scale isn't even remotely comparable. The first explorers on their clumsy ships could traverse the Atlantic in a few months. Te likely requirements for intersteller travel are decades to centuries. But that's not even the whole story - Sailing ships can be powered as they go. When they get within sight of land they just furl their sails and they stop. Space travel is a completely different set of physics. First you have to expend an enormous amount of energy to leave your own solar system. Then around mid-way,the gravitational pull of the destination star starts to accelerate towards your destination. By the time you reach the new system you are traveling faster than when you left, requiring even more fuel to stop than you needed to leave your home system. Returning only compounds the problem, as you'll need yet more energy to leave and then slow down again. Put another way, only about 10% of your energy budget is spent leaving your solar system for a round-trip voyage.
3) there's nothing in the middle, and the middle is really, really big. Sailors crossing the oceans carried stores with them, but they weren't completely self sufficient. Wind still powered their crossing, rain replenished their drinking water and they didn't need to worry about oxygen or heat. Virtually all voyages relied on periodically replenishing their food stores with fish and with soirées onto various islands. All of these can't happen in deep space. I'll admit this is just a technological hurdle, and an advanced society certainly could build a completely self contained space-ship with its own centuries-long power source, but what I still don't see is how this would be economical. I'm not saying it can't be done, just that it is so costly that it might never be attempted.
Edit: While I'm still doubtful, I'll admit that the possibility exists, particularly for life-forms quite different from ours. Specifically, I think the 'ideal' colonization candidate would be a species with a vastly longer lifespan (perhaps thousands of years), are consideribly smaller than humans (requiring less fuel between stars) and are tolerant of a wide variety of physiological conditions, including cosmic rays. Intelligent cockroaches perhaps, or better yet sentient microbial life.