About "builing the ship"
Any civilization that can build ships that can travel at close to lightspeed has other advanced technologies at its disposal - for one, advanced nanotechnology.
FOR US, building a colony ship would be a huge drain on world economy; NOT for a civilization that mastered nanotechnology and has at its disposal the resources of an entire solar system.
Motivations for colonizing other places are numerous (as our own history showed) - population pressure; the desire to escape oppression; in order to start "fresh" and not be just a cog in the huge societal machine; certain cultures will wish to be "independent", etc.
The point is - any alien species will be composed of many different cultures with different motivations; some of these civilizations will have the motivation to "move out". And once colonies are established amd develop their own culture, the number of different cultures/motivations will increase (for example, since 1500 until today - aka in 500 years - humanity changed enormously/increased in numbers at an impressive rate) - some of these colonies will soon send ships further out, some not so soon. Per total, as the number of colonies increase, the number of colonizing ships will increase.
About the "dangers" and "speed" of space travel
Outer space, unlike what we see in Star Trek is BORING. REALLY BORING.
Essentially, space is a huge, empty desert with few resources of any kind (and the solar systems are "oasis"). The greatest danger in outer space is to remain without resources. The best way to minimize this danger is to travel at yout top speed - aka near light-speed - once you have decided where you want to go - and, with the FTL speed limit, you WILL NOT LEAVE WITHOUT DECIDING your destination.
Yes, it's possible some intelligent species will never develop technology - as far as SETI is concerned these species are always silent. It's also possible that some intelligent species will not have the desire to explore (which is NOT an evolutionary advantage - humanity, without the desire to explore/expand would be stuck in Africa/most likely extinct) - of course, I don't see how these species become dominant on their planet/develop technology.
But, among the alien species - all with different motivations - there will BE some with the ability to create technology/the desire to expand (a definite evolutionary advantage, as said). I have been talking about these species. It only takes ONE SUCH SPECIES and a few MILLION YEARS to colonize the entire galaxy.