To demonstrate the philosophy behind RK methods, let us consider the second-order RK method, RK2. The first approximation consists in Taylor expanding $f(t,y)$ around the center of the integration interval $t_i$ to $t_{i+1}$, that is, at $t_i+h/2$, $h$ being the step. Using the midpoint formula for an integral, defining $y(t_i+h/2) = y_{i+1/2}$ and $t_i+h/2 = t_{i+1/2}$, we obtain $$\begin{equation} \int_{t_i}^{t_{i+1}} f(t,y) dt \approx hf(t_{i+1/2},y_{i+1/2}) +O(h^3). \tag{37} \end{equation}$$ This means in turn that we have $$\begin{equation} y_{i+1}=y_i + hf(t_{i+1/2},y_{i+1/2}) +O(h^3). \tag{38} \end{equation}$$