Question: I read somewhere that when the aircraft banks to turn, its angle of attack increases.Since angle of attack is the angle between the relative wind and chord line, how does turning increase it? Is it because the relative wind is now from different direction or is it because the wing that goes higher meets the air at a higher angle?

Answer: 
Just to avoid some confusion that I feel in your question:

It doesn't!
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That is, it doesn't happen by itself. The pilot or autopilot must actively pitch up (and possibly do other adjustments, e.g. increase thrust). Without active control, the angle of attack will stay about the same (initially), and the airplane will start descending if it enters a turn, because the lift becomes insufficient for level flight (see pictures in other answers).
The question itself is open ended but, in the simplest terms, it depends on how you execute the turn - are you flying a constant airspeed or a constant angle of attack?

In either instance, in a level turn, the load factor will increase with the inverse cosine of the bank angle.

If flying a constant airspeed turn, then an increase in lift coefficient will command a higher angle of attack as dynamic pressure remains constant: CL-turn = CL-wings level (L-turn/L-wings level)

If flying a constant angle of attack turn (i.e., same angle of attack in the bank as that used in level flight), the dynamic pressure has to increase to support the higher wing loading at a constant lift coefficient. Since dynamic pressure is a function of the velocity squared, the velocity commanded to maintain level flight within a banking turn varies with the square root of the load factor: V-turn = V-wings level x (load factor)^1/2 ...all other factors being equal.