The excerpt was from a much longer quote, and he's completely right...while we will no doubt make use of our solar system, it makes much more sense to "seed" the galaxy for our survival
Seeding the galaxy is not NECESSARY for our survival, thus doing so for survival's sake makes no sense. It would be like moving to a desert island to avoid your neighbor's dog.
Hawking is incorrect. The solar system bodies and their accessible resources are not only a viable choice, they are also a logical first step that would precede any attempt at interstellar transportation anyway. Thus, by the time we are in any position to BEGIN to colonize beyond the solar system, we will no longer NEED to.
You could also be a little less conservative and the seeding of other galaxies wouldn't even take much more time than the galaxy...
On the timeline and scale we're talking about, it would actually be more feasible to refuel the sun than to even ATTEMPT to colonize other galaxies.