Certainly there are attractive interactions between water molecules and nonpolar solutes, there are always dispersion attractions. Additionally, water will "layer" against a solute particle, producing solvation shells, even in the complete absence of attractions, just because of the excluded volume forces packing everything together. However, for nonpolar gases the new gas-water interactions are significantly weaker than the water-water interactions they replace, so the enthalpic change on dissolve is strongly positive (unfavorable). Hence the only reason gases dissolve is because the entropy increases, and that is simply because the volume available to the gas molecules is larger if it includes the volume of the water as well as the volume of the gas above it.
Another way to put that is that molecules of nonpolar gas are found in the water simply because they can be. Id est, those molecules will zoom around in the gas phase, occasionally strike the surface of the water, plunge in, and wander around randomly for a while before exiting. Net result: a low concentration of nonpolar gas dissolved in the water. The concentration will increase with pressure simply because a higher pressure means a higher density of the gas, which means more gas molecules will strike each cm2 of the water surface per second, and at equilibrium that must be matched by a higher rate of exit from the surface, which requires a higher equilibrium concentration in the bulk water. For small pressures, the effect is approximately linear in pressure, and the coefficient is called the Henry's Law coefficient.
The dissolved gas particles do not occupy spaces between the water molecules, if by that you mean preexisting spaces. They shove the water molecules out of the way, create a hole, and live in that. The volume of the solution may not increase, however, or may not increase as much as one might think, because as I mentioned above, the water will form solvation shells around the dissolved particle, and will moreover disrupt the hydrogen-bonding network in the water. Both effects could result in an increase in density in the regions near the hole created by the solute particle. So predicting whether and by how much the volume of the water expands (or contracts) is in general difficult.