When you apply pressure to an incompressible fluid the pressure is transferred. If you stop applying the pressure there is no motion. Volume and density are integral parts of calculating how much energy is required to achieve certain useful motion like lift a truck or press a part. Why don't certain fluids compress and others do? If you start with a large pressure on one end of a fluid tube and you have an even larger pressure on the other end of the tube will the fluid move the moment you increase the pressure to a slightly higher level? What is the reason water doesn't compress or move like air?
Fluids like water do compress, though in a fluid the bulk modulus is much higher than in a gas. The same applies to solids.
In a gas the pressure is mostly due to the motion of the gas molecules, while in fluids and solids the bulk modulus is mostly due to intermolecular/interatomic forces, so the resistance to compression comes from a completely different mechanism
The diagram above shows roughly what the force between two atoms looks like as a function of distance. If you're interested I grabbed the diagram from here. In liquids and solids the interatomic distance is about at the minimum of the force, and that means if you try and push the atoms closer together you hit the "hard core" i.e. the force increases very quickly. That's why it's hard to compress liquids and solids. In gases the interatomic spacing is far to the right on the graphs so you can compress the gas a long way before you hit the hard core.
With fluids that contain large molecules, like oil, the potential is a more complicated because the molecules in the fluid can change their shape under pressure, but the general behaviour is the same.
Re your question about transmitting changes in pressure: in a fluid (or solid) pressure changes are transmitted at the speed of sound. So if you suddenly increase the pressure in your hydraulic tube the increase in pressure takes a finite (but small) time to reach the other end of the tube.
Browse on Pascal's Principle.
The incompressibility of a fluid (liquid or gas) depends on the structure of its molecules. Thus, gases have higher compressibilities compared to liquids. This is because gases have more dispersed particles than liquids. This is analogous to a big soccer field with a lot of people standing far away from each other. Liquids are more compressed than gases.
Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid, as well as on the walls of the container. Therefore, pressure remains unchanged, but force on the particles of the fluid do, depending on the cross-sectional area of your container.