The equation is a deterministic equality for every measurable bounded function B:[0,1]\to R by a theorem on existence of the Stieltjes integral, since \psi_n is of bounded variation. For simple explanation see e.g.
Thus, there is no need to refere to Ito stochastic integration. To the contrary, I think that the picture presents an introduction to the integral developed on next pages, which is supposed to be obvious - at least to those familiar with Rieman-Stieltjes integral. The formula could be seen as an application of a deterministic integral wrt to random function (more precisely - it is defined almost surely).