Why does paint stick to wall?

There are three main ingredients in any paint:

1. solvent ("carrier"),

2. color,

3. and film-former ("resin" or polymer).

Solvents are liquids (alcohol, toluene, water, etc.) which carry the color. Nowadays, most paints are water-based, meaning that their solvent is mainly water. Color is added with either dye or pigment. More often pigment is used because it lasts longer. Finally, resins are added which give the paint various characteristics, including how the paint flows, dries, how smooth it looks, how it sticks to different surfaces, etc. Resins are polymers which come in all shapes and sizes; some are natural like gums or starches, and a lot are synthetic like acrylic polymers used in house paints. Other stuff that is added to paints: fillers like clay or opacifiers (make the paint hard to see through), surfactants (chemicals that help stabilize the paint so it lasts longer or brushes easier), "antimicrobial" agents (keeps bacteria and bugs from growing in water-based paint), etc.

By pushing the brush to the wall you apply a force. The paint is pushed to the wall and will be in touch with the wall. Due to the resin, the paint has a high visco-elasticity, which means that it can be streched without breaking (like bubble-gum). By pushing the paint to the wall and move the brush along the wall, you 'strech' and 'scratch' the paint from the brush onto the wall. Surface tension also plays a role: liquids, like the water used in paints, always want to minimize their surface, hence want to form droplets. However, the paint is pushed onto the wall and needs extra paint from the surrounding area to form droplets. So the paint on the wall attracts extra paint(water) from the brush to be able to form droplets on the wall.

In regards to how paints dry on a molecular level: imagine that the resin exists as tiny drops in the water surrounded by tiny surfactant molecules. Also imagine the pigment as tiny (particles) in the water surrounded by surfactants... Once you brush paint out onto the wall, the resin droplets begin to lay down onto the surface as the water or solvent begins to evaporate. The resin droplets then "coalesce" or migrate closer together and bind (stick) to each other. The pigment particles begin to stick to the resin. The resin droplets eventually fuse together into one long polymer film on the wall, with the pigment particles trapped and stuck in it.