Here is my two cents on an intuitive explanation of the Ito integral:
The Ito integral is $\int_S^T f(t,w) dB(t,w)$
We can thing of $B(t,w)$, the Brownian motion as the actual price (with mean subtracted) and $f(t,w)$ is a random trading action and its gain on the observable prices. As a result, $f(t,w)$ is $F_t$ adaptive, i.e., it can be dependent only on the history of the prices not future prices. Then, the Ito integration is the total gain from $S$ to $T$ using random trading action+gain $f(t,w)$.
