I assume they are using non-linear to mean non-continuous, which implies that there can be large, hard-to-understand changes in behavior when the input is changed only a small amount.
Polynomials with large degrees are continuous. It's just that they can still change by a large amount (i.e. having a large derivative) when the input is changed by a small amount.
I invite you to construct the Lagrange polynomial (i.e. interpolating polynomial) for points on a nice, simple curve with some noise. They will, by definition, pass through every point given, and yet it will likely behave very badly outside the range of the given points.
There is nothing wrong with using a non-linear model, though; x^2 or x^3 regressions make sense on many datasets.
Non-continuous is also not the perfect terminology, but I argue that it is more precise than non-linear: the chief idea being that the model "changes unpredictably."
I invite you to construct the Lagrange polynomial (i.e. interpolating polynomial) for points on a nice, simple curve with some noise. They will, by definition, pass through every point given, and yet it will likely behave very badly outside the range of the given points.