This method can be derived by writing two Taylor series about a fictional grid point at $(n+\frac{1}{2},\, i)$, evaluated at point $n$, and point $n+1$.
$$ f_i^n = f_i^{n+1/2} - \frac{\Delta t}{2}\left.\frac{\partial f}{\partial t}\right|^{n+1/2}_i + \frac{\Delta t^2}{4}\left.\frac{\partial^2f}{\partial t^2}\right|_i^{n+1/2} + \mathcal{O}(\Delta t^3),$$$$ f_i^{n+1} = f_i^{n+1/2} + \frac{\Delta t}{2}\left.\frac{\partial f}{\partial t}\right|^{n+1/2}_i + \frac{\Delta t^2}{4}\left.\frac{\partial^2f}{\partial t^2}\right|_i^{n+1/2} + \mathcal{O}(\Delta t^3),$$Subtract the first equation from the second: $$ f_i^{n+1} - f_i^n = \Delta t\left.\frac{\partial f}{\partial t}\right|_i^{n+1/2} + \mathcal{O}(\Delta t^3).$$
Now, we take $$\left.\frac{\partial f}{\partial t}\right|_i^{n+1/2} = \frac{1}{2}\left(\left.\frac{\partial f}{\partial t}\right|_i^n + \left.\frac{\partial f}{\partial t}\right|_i^{n+1}\right) = \frac{1}{2}\frac{\alpha}{\Delta x^2}[(f_{i-1}^n - 2f_i^n + f_{i+1}^n)+(f_{i-1}^{n+1} - 2f_i^{n+1} + f_{i+1}^{n+1})].$$
Inserting gives $$f_i^{n+1} - f_i^n = \frac{\Delta t}{2}\frac{\alpha}{\Delta x^2}[(f_{i-1}^n - 2f_i^n + f_{i+1}^n)+(f_{i-1}^{n+1} - 2f_i^{n+1} + f_{i+1}^{n+1})],$$
which rearranges to the final result.