Calculating $\prod\limits_{k=0}^\infty(1+{\rm e}^{-(2k+1)\pi})$

Question

Someone computed it by Mathematica and got that $$\prod_{k=0}^\infty(1+{\rm e}^{-(2k+1)\pi})=2^{\frac14}{\rm e}^{-\frac\pi{24}}.$$ However, I don’t know how to prove it.

I would appreciate it if someone could show some suggestions. Thank you.