Is space expansion causing increasing distances between galaxies? I read/heard that the expansion of the universe would lead to any observer not being able to see other galaxies as time goes on, due to their light not reaching them anymore.
But there's something I quiet can't figure about that.
I always understood space expansion as an homothety (because space expands evenly?). Is that correct?
If so, aren't all ratio of distances to their image supposed to be constant after a scalling transformation?
If the "ruler" we use to mesure distances between two galaxies expands with its space, how does the distance that light has to travel to get to us also increases?
 A: 
I read/heard that the expansion of the   universe would lead to any observer not being able to see other galaxies as time goes on, due to their light not reaching them anymore.

This is correct. Space expands and galaxies travel away from each other, carried along by the expansion (called the Hubble flow). There are presumably galaxies already outside the observable Universe, as the light has not had time to reach us yet.
If the expansion continues (and continues to accelerate, as we believe it is doing due to dark energy), eventually galaxies will become completely isolated in space, with the distances so enormous that light will not be able to traverse it.

I always understood space expansion as an homothety (because space expands evenly ?). Is that correct ? 

I think you mean homogeneously, and yes, that is broadly correct. Gravity is generally locally strong enough to overcome the expansion though, which is why galaxies can form in the first place. In those areas, the expansion is not technically homogeneous, as gravity is working against it by different amounts in different places.

If so, aren't all ratio of distances to their image supposed to be constant after a scalling transformation ?

In cosmology, we do like to define what is called the cosmic scale factor, which is equal to 1 today and 0 (or very close to it) at the beginning of spacetime. This helps us to understand how physical distances have changed over time, but it doesn't mean that those physical distances don't change.

If the "ruler" we use to mesure distances between two galaxies expands with its space, how does the distance that light has to travel to get to us also increases ? 

The physical separation between non-gravitationally bound objects (eg distant galaxies) is growing. You can define a scale factor that changes with your expansion so that the distance looks like it remains constant. But again, I stress the physical expansion is still real.
Draw two dots close together on a balloon and then blow it up. See how the dots move apart as "spacetime" i.e. the balloon's surface expands. That is basically what is happening in the Universe.
A: 
I always understood space expansion as an homothety (because space expands evenly ?). Is that correct ?

It's a homothety of the three-dimensional spatial geometry, but what we're dealing with in general relativity is the geometry of the four-dimensional spacetime. The notion of a three-dimensional spatial geometry only makes sense if you first decide on what you mean by a surface of simultaneity. Such a surface does have a fairly natural definition in the special case of a homogeneous cosmological spacetime, but it's not something that is built in to the structure of general relativity.

If so, aren't all ratio of distances to their image supposed to be constant after a scalling transformation ?

Yes, but when you observe light from a distant galaxy you are not just measuring a distance. General relativity does not really define the spatial distance between two points. When people talk about the distance between two galaxies, and those people understand how GR works, they know that they are talking about a certain mathematical construct that only makes sense because of certain special properties of cosmological spacetimes (basically the existence of a preferred local frame associated with the Hubble flow).
When you observe light from a distant galaxy, you are making an observation that depends on all the details of how the light traveled through the expanding universe. This motion took place over some long period of time, during which the geometry of the universe was changing dynamically.

If the "ruler" we use to mesure distances between two galaxies expands with its space, how does the distance that light has to travel to get to us also increases ?

Rulers do not expand by any significant amount due to cosmological expansion. If they did, then cosmological expansion would not be observable. (The part about "by any significant amount" is necessary because there are theoretically very tiny strains on a ruler due to the acceleration or deceleration of cosmological expansion, but these are much too small to measure in practice and do not necessarily show any secular trend.)
A: The galaxies drift apart as time goes by because in their free-fall motion they are on trajectories with direction and velocity which makes them get further apart as time goes on.
The cosmic expansion is precisely that.
There is no universal expansion of all distances; that is not what the cosmic expansion means. Each galaxy is in free-fall, and as it falls it is bound together by its own internal gravitational forces, so it does not get any bigger. Planets and stars are even more tightly bound by electromagnetic and related forces, so they don't get any bigger either.
The notion of 'space expansion' arises because the Newtonian picture of space fails, and in the General Relativistic picture one can talk about an unbounded space with finite volume, and the volume can get larger as time goes on. That is one model they may possibly match our universe. In this model, as
the galaxies (actually, galaxy clusters) get wider apart the available cosmic volume increases at the same rate, so the galaxies never move relative to the cosmic sphere since the sphere just gets larger as they get further apart.
Other possibilities for the global geometry (flat, or curved the other way) are less easy to visualize but similar considerations apply.
