Why is a wavelet transform implemented as a filter bank? The mother wavelet function $\psi(t)$ must satisfy the following:

$$\int\limits_{-\infty}^{+\infty}  \frac{|\psi(\omega)|^2}{\omega} d \omega &lt; +\infty,$$ 
$$\psi ( \omega ) \bigg|_{ \omega =0} =0,$$
and 
$$ \int\limits_{-\infty}^{+\infty}  \psi(t) \ dt = 0$$

To serve as the wavelet basis for wavelet transform 
$$ \gamma (s, \tau ) = \int\limits_{-\infty}^{+\infty} f(t) \ \psi_{s, \tau }(t) \ dt$$

where $ \psi_{s, \tau }(t) \triangleq \psi\left( \frac{t-\tau}{s} \right)$.

While I understand that the wavelet must be an oscillatory function having no frequency component at $\omega=0$ and effectively have a band pass filter like spectrum, from the equation of wavelet series or wavelet transform can you tell me why is it that the wavelet transform is implemented as a filter bank? What is the intuition behind it? What makes it possible?

I am asking this question since the fact that practically the DWT is implemented as a filter bank means that it is not a DWT anymore, it is just a set of low pass and high pass filters. It is mind bogling.
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