Why is a wavelet transform implemented as a filter bank?

The mother wavelet function $\psi(t)$ must satisfy the following:

$$\int\limits_{-\infty}^{+\infty} \frac{|\psi(\omega)|^2}{\omega} d \omega < +\infty,$$ $$\psi ( \omega ) \bigg|_{ \omega =0} =0,$$ 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.