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 AoA in a turn will be higher than in level flight IF the same airspeed and level is maintained. Reason being, in a turn some of the "upwards" (relative to the lateral axis) force generated by the wing is used to turn the aircraft around, so more lift needs to be generated to maintain altitude. To generate more lift at the same airspeed, the angle of attack has to increase.

This image illustrates it nicely:
[![http://avstop.com/ac/flighttrainghandbook/imagefvn.jpg](https://i.stack.imgur.com/2fDJ5.jpg)](https://i.stack.imgur.com/2fDJ5.jpg)  

source: <http://avstop.com/ac/flighttrainghandbook/forces.html>

(consider the "Total Lift" vs. the "Vertical Component of Lift")