Why is a wavelet transform implemented as a filter bank?
First of all the basic idea of wavelet transforms lies in multi-resolution analysis. What this means is that the signal is looked at from different scales.

It is probably easier to understand this with images (which are 2D signals). The idea of multi-resolution is like zooming in and out of a reference signal and compared with windows of the image according to the amount of zoom.

Similarly for 1D you are trying to see how the mother wavelet at different scales (think of it as compressing and expanding in the 1D) compares with the signal at different delays (at different points on the discrete 1D). 

Now when you look at a given set of discrete points you want to transform based on a mother wavelet it easy to see them as filters (based on mother wavelets) that are applied at different scales.

For a better understanding of wavelets and DWT I suggest you read Polikar's tutorial : Wavelet Tutorial