Proving $\prod_{n=0}^{\infty}\left(1+\frac{x}{a^n}\right)=\sum_{n=0}^{\infty}\frac{(ax)^n}{\prod_{k=1}^{n}(a^k-1)}$
Solution 1:
For $|a|>1$ let $$f(x)=\prod_{n=0}^\infty (1+x a^{-n}) =\sum_{m=0}^\infty c_m x^m$$ $$\sum_{m=0}^\infty c_m a^m x^m=f(ax)= (1+x a) f(x)=1+\sum_{m=1}^\infty (c_m+a c_{m-1})x^m$$ Equating the coefficients we find that $$c_0=1,\qquad c_m a^m=c_m +a c_{m-1}$$ ie. $$c_m = \frac{a}{a^m-1}c_{m-1} = \frac{a^m}{\prod_{k=1}^m (a^k-1)}$$
Solution 2:
In accordance with OP’s request, I am posting my comment below.
The identity is a theorem of Euler’s that comes from the $q$-binomial theorem in the limit $n\to\infty$. A proof can be found here.