To understand the process you should first realize that air, and water, just like all matter, are comprised of atoms and molecules, with a great deal of empty space between them. In the case of air, it's about 78% nitrogen (N2 molecules), 21% oxygen (O2 molecules), 0.9% argon (Ar atoms), and small fractions of a percent of carbon di-oxide (CO2), etc. (see http://www.physlink.com/reference/AirComposition.cfm) Water is of course comprised of molecules made up of two hydrogen atoms and one oxygen atom each (H2O).

When water evaporates, some H2O molecules gain enough energy to leave the liquid phase and become gaseous. These molecules can now move between the atoms and molecules comprising the air above the liquid. However, as these atoms and molecules move around, they interact (in simplistic terms, they collide and bounce off each other). Since there is nothing holding the atoms and molecules in a particular volume, having more of them in a limited volume increases the density of the air. Over time (on a scale of seconds or minutes), the excess water molecules in the nearby volume begin migrating, gradually dissipating over an ever-greater volume.

So in the simplest of terms, the answer to your question is that the water becomes "contained" in the air, and then gradually the excess dissipates to a gradually increasing volume reducing local humidity back to its original level. In a different scenario, such as a sudden breakage of a high-pressure steam pipe, the steam (gaseous water) displaces, or pushes away the air. Once the flow of steam stops, air will mix back in and the same gradual return to normal will occur.