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).