Why do hygroscopic compounds absorb water from the air?

Hygroscopic compounds take up water from the air for many different reasons. Here are a few:
Ionic compounds are often hygroscopic because they form stable hydrates.
Metal cations (being positively charged) attract the lone pairs on water oxygens and form coordinate covalent bonds with water. For example, many divalent cations M2+ can form ions like [M(H2O)6]2+. One or more of the waters can be replaced with anions in the hydrate.
Anions like sulfate or nitrate can form strong hydrogen bonds with the water. For example, nitrate ions form a very stable arrangement with water in which the water "bridges" the nitrate oxygens with hydrogen bonds.
Sometimes both the anion and the cation bind water. For example, in copper(II) sulfate pentahydrate, four waters are bound in a square around the copper(II) ion's equator and two sulfates are bound to the poles. The fifth water bridges sulfates together.
Some highly soluble ionic compounds go a step further. The shell of waters formed around the individual ions on the surface of the material becomes an excellent surface for more waters to bind. This is especially true with small and highly charged ions like hydroxide. Water will sometimes continue to be added to the shell until the material actually dissolves in the adsorbed water. This is called deliquescence. As a student, I remember naively trying to prepare an 0.1000 M NaOH solution by weighing out a tenth of a mole of NaOH on a Mettler balance. I couldn't get a steady weight; the balance was precise enough to weigh the water as it was being absorbed from the air. Within a few minutes, I could see that the surface of the pellets was wet, and within about 15 minutes, the pellets looked as though they had melted.
Water molecules can diffuse into the material and become trapped there, especially if the material contains voids that can hydrogen-bond the water in place.
Some porous materials can absorb large amounts of water by a phenomena called capillary condensation. For example, zeolites (complex sodium aluminum silicates) are honeycombed with cavities lined with oxygen atoms. Water in air hydrogen bonds to the outer surfaces of the material. As more water condenses, it tries to spread out and wet as much of the surface as possible to minimize its energy through hydrogen-bonding. That carries the water deeper into the material, making more room for water to adsorb to the outer surface. (They soak up water like a sponge!)