Someone else’s take on this topic.

If aliens could travel at a fraction of the speed of light, why haven’t they colonized our Galaxy by now?

We’ve all been brainwashed by years of Star Wars, Star Trek, Marvel Universe, Avatar, etc, to think that space should be teeming with intelligent civilizations, most of them vaguely like ourselves, working with and against each other to carve up the galaxy. As a result, it’s easy to overlook the huge assumptions embedded in your question.

• Habitable worlds exist. Do they? It seems overwhelmingly likely, given that there are probably a trillion planets in the Milky Way alone, but for now we don’t know. Perhaps there are many near-miss planets like Venus and Mars, but extremely few true Earth analogs. For instance, life might require a particular rock/ice ratio, a large moon, and a specific style of plate tectonics. That level of specificity seems unlikely to me, but that’s just my random opinion. Until we find another planet with truly Earthlike conditions, we cannot say for sure that this is true.
• Alien life exists. Does it? Honestly, we have no idea. There are many strong arguments suggesting that the fundamental biochemistry of self-replication is practically inevitable given the right conditions. But we don’t know how common those conditions are (see above), and even then we don’t know if there is some extremely low-probability gap that hinders the emergence of even simple microbial life.
• Intelligent life exists. Does it? This one is a complete unknown. Keep in mind that there was no intelligent, self-aware life on Earth for 99.999% of its existence. Maybe the emergence of intelligence here was a rare fluke, unlikely to be reproduced anywhere else. Rat-level intelligence seems to have existed for at least 200 million years without any indication that higher level intelligence would confer a big evolutionary advantage. (There are all kinds of speculations about why intelligent life could not emerge until now on Earth, but these are just-so stories, trying to paint an explanation on top of a truth that we already know.)
• Intelligent species want to “colonize” the galaxy. Do they? Life does have a tendency to explore every available ecological niche, and humans sure do like to spread out. From our example of one Earth, it seems likely that this is a general tendency of life everywhere, but we are doing an awful lot of extrapolating here. Maybe other types of intelligence have other motivations that have nothing to do with expansion.
• Intelligent species become technological species. Do they? It’s certainly true for humans, but dolphins have a high level of intelligence and they are not trying to build spaceships. Crows, chimps, and bonobos are also capable of simple tool use, but they don’t appear to have experienced any evolutionary pressure to become true technological species.
• Technological species can travel a significant fraction of the speed of light. (I assume you mean something like more than 1% of light speed.) Can they? Extrapolating from human technology, that seems extremely likely. Then again, the fastest spacecraft we have ever built would take about 300,000 years to reach the next star. Nobody is going to be colonizing the galaxy at that rate. You have to accept that speculative but unproven technologies are both feasible and practical for more advanced technological civilizations. Maybe intelligent life is out there, but in isolated pockets.
• Intelligent, technological, space-faring species survive for a long time. Do they? Oh boy, we have no idea at all if this is true. Earth is 4.5 billion years old. Life has been around 4 billion years. Land species have been around 400 million years. Rat-level intelligence has maybe been around 200 million years. Our species has been around for about 100 thousand years. We have been capable of spaceflight for less than 100 years. It may seem inconceivable that humans could go extinct—but even if we last another 100,000 years, that may not be nearly enough time to spread across the galaxy, even if we develop the means to do it and maintain the will to do it. If intelligent species typically last less than 100,000 years, thousands of them could have come and gone in our galaxy without us ever knowing.

So there’s not one answer, but a whole set of overlapping possible answers why we don’t see evidence of any alien civilizations around us. And that doesn’t even consider more exotic possibilities, such as the idea that they might be here but just undetectable to us or deliberately hidden from our primitive eyes.