The water molecules are attracted to the surface of the structure by adhesive forces. This means that when the water molecules are away from the structure, there is "potential adhesive energy". As the water moves closer to the surface, the "potential adhesive energy" is reduced and turns into gravitational potential energy -- until at some point an equilibrium is reached. The more surface area the liquid can adhere to, the higher it can rise.
If this is hard to understand, think of two strong magnets, one lying on a table, and one mounted at a point somewhere above the table. When the two magnets are not touching, there is potential magnetic energy. If you bring the top magnet low enough, at some point the bottom magnet will be pulled up towards the other magnet until it touches, increasing it's potential gravitational energy and minimizing its potential magnetic energy.
Kinda guessing, but I'd assume from temperature - i.e. chaotic (a.k.a. brownian) motions of the particles resulting in them randomly getting close enough to the enclosure's walls that the surface tension (which from what I see on Wikipedia seems to be basically electric attraction of particles) "glues" them to the wall, and/or to particles that got already "glued" to the wall just moments before. I would suppose "in real life" the resulting cooling is probably basically irrelevant and gets immediately "replenished" from the surroundings, and (again also from what Wikipedia seems to say) the equilibrium seems to be reached vs. gravity, presumably mostly of the "non-glued to walls" part of the water column. I.e. I guess when the freeriding bunch of suckers in the middle of the column pull you down more than the sweet, sexy wall pulls you up... errr, I mean, the sum of the gravity force acting on the water molecules in the middle of the water column equals the sum of the wall-sticking force of all the wall-sticking water molecules (distributed locally through the water-to-water-sticking force of the water molecules).
The energy comes from the work done when the surface of water was broken, plus the already existing kinetic energy of the water, and the localized energy in the wall of tube. (Basically getting the surface of the tube wet is energetically more favorable than not, so this means there's a hidden/latent energy there.)
My highly uneducated guess on this would be that the water molecules moving up are cooling down a tiny little bit.
Being cooler would then be making it unable for them to climb further when a certain threshold is reached, meaning they would need the surrounding environment to heat it up a little again to allow it to move further.
