In my symmetries of classical mechanics course we have looked at taylor expansions. Our notes claim that; $$ f(x + a) = \sum_{n=0}^\infty \frac{1}{n!} f^{(n)}(x)a^n ≡ \exp{\left( a \frac{d}{dx}\right)} f(x) $$ I am happy with the first equality, its explained quite nicely in this question. The second equality is puzzling though. I dont even know what to make of the derivative inside the exponent.
How is the second equality derived?
It applies the exponential function $\exp(y) = \sum \frac{1}{n!}y^n$ to the differential operator $a\frac{d}{dx}$ to give $$\exp\left(a\frac{d}{dx}\right) = \sum \frac{1}{n!}a^n \frac{d^n}{dx^n}.$$ This, when applied to $f(x)$, gives your middle expression.