The cheap and easy answer to this is that the double pendulum is considered chaotic because it is very sensitive to small perturbations in initial conditions (amongst other things). Showing this mathematically may be difficult (see the Lagrangian formulation for the dynamics), but if one looks at the animations on the Wikipedia page showing the trajectory of the double pendulum, the intuitive reason for this sensitivity should become obvious. There are many points in the trajectory where the acceleration rate of the outer pendulum is very dependent on the exact angle of the upper pendulum as it is whipped around. If the inner pendulum is in a sightly different place, the outer pendulum is whipped around at a very different rate, changing how "coupled" the two pendulums are. Sometimes the effect is to tie them together like they were a string on a grandfather clock. Sometimes it causes them to be almost perfectly opposed in position, doing their own thing.
Every time it reaches one of these states, it becomes very sensitive to the initial conditions that lead it to that state. A sight perturbation along the way could have arbitrarily magnified effects later.