While it's true that space is expanding, this is a grossly oversimplified representation of the phenomenon.
We don't yet fully understand the exact reason and mechanics of how the universe is expanding, or how it will behave in the distant future. However, what we do know is that this expansion rate is not uniformly distributed throughout space and matter; that is to say, individual atoms are not getting farther apart from each other. Why would galaxies grow more distant from one another, yet retain their internal structure and composition? The answer is that other forces take predominance over the expansion.
This isn't an unfamiliar principle in physics; many forces have ranges within which they are more or less powerful. For example, the strong force is more powerful than electromagnetism, but it can only act over a short range. The current model appears to show that the forces driving the expansion of the universe are either less powerful, or operating on a different range, than gravity.
We are part of the Virgo Supercluster, a grouping of thousands of galaxies, all of which are interacting through gravity. While the distance between the VC and other superclusters is expanding, all of the contents of the VC are being predominately attracted to each other. Thus, it is entirely possible for galactic collisions to occur, regardless of the expansion of the universe.
The answer to the original question is, frankly, luck. Modeling galactic interactions based on gravity takes place on a scale of millions of years and does not necessarily yield observable results; as Saumya-Dave stated, our current models anticipate the Milky Way and Andromeda galaxies colliding in several billion years. So, in fact, the question is not "why haven't we collided" but "why haven't they collided YET".
Galaxies are moving but there is a theory that in the future, the Andromeda galaxy will collide with our galaxy. Meanwhile, galaxies are colliding with each other with is really rare, but they then create a new galaxy even though they can pass though each other safely, the crashing of stars, planets and asteroids is super rare.
Yeah! our galaxies and other galxies are also moving but don't forget our cosmos is also expanding, the distance between two galaxies is increasing day by day.
This was first discovered by Edwin Hubble by finding the frequencies of the light rays. As the frequencies were becoming low, it was discovered that the universe was expanding.
Yeah I even heard of that.
The galaxies are not just moving, they are moving away from each other. This can be understood simply by taking an example of a balloon which has spots marked over its outer surface. As we blow the balloon, every spot moves away from other spot. Thus, they never "collide" or meet. Same is the case with galaxies. Edwin Hubble discovered that galaxies are moving away from one another. Thus, instead of colliding, one day every galaxy will be very far away from other galaxies and in general terms, every galaxy will be very alone. So there will be little probability of further galaxies colliding.