Look up something on [Tikhonov regularization](https://en.wikipedia.org/wiki/Tikhonov_regularization), also known as *ridge regression* in machine learning. This is a standard technique (but I agree that the explanation in that notebook is somewhat poor).

Technically speaking, it does not affect the *numerical stability* of that algorithm, but it modifies the problem to a more *well-conditioned* one, from $\min \|\Phi \theta - y\|^2$ to $$\min \|\Phi \theta - y\|^2 + \kappa \|y\|^2.$$