Let’s just imagine for a moment that humanity eventually advances in technology such that light-speed travel is attainable. The speed of light, roughly 3×10^8 m/s, is the upper limit on travel speed- If light could go any faster, it would.
With light-speed travel, humanity will likely attempt to colonize other planets, for a variety of reasons- curiosity, resource depletion, or even a sense of manifest destiny. Imagine, for a moment, that you are hoping to travel to a habitable planet in the Andromeda galaxy. Andromeda is 2 million light years away. This means that, even if a method of light-speed travel is devised, it would take 2 million years to reach a planet in one of the galaxies nearest to us. For reference, humans have been around for about two hundred thousand years.
Even with an endlessly vast universe, the chances of humans leaving the Milky Way seem to be essentially zero.