In my stochastic course, we defined the stochastic integral for $h \in \mathcal L^2$ (approximating with simple processes) : $$\int_0^T h_s dW_s$$
Now we want to do the same for $h \in \mathcal L$ but I do not understand the construction we are doing this time : it is done using "localization".
We define the stopping times $$\tau_n =\inf \{t | \int_0^t\lvert h_s \rvert^2ds \geq n \} \wedge T$$ then let $h^n := h \mathbb 1_{[0,\tau_n]} \in \mathcal L^2$. My course then says we can define unambiguously $$\int_0^T h_s dW_s = \lim_n \int_0^T h_s^n dW_s $$ and $$\int_0^T h_s dW_s = \int_0^T h_s^n dW_s = \int_0^T h_s^{n+1} dW_s \text{ on } \{ \int_0^T \lvert h_s \rvert^2 ds < n\} $$
I really do not understand how this definition works. Can someone make it clear ?