This is just a formal proof using idea of Loïc Teyssier that appeared in the comments. We just rewrite both sides of the equation as double sums (using geometric series) and notice that they are equal after changing the order of summation.

Since $$\frac{q}{1-q} = \sum_{k=1}^\infty q^k,$$ we have for the left hand side

\begin{align}
- \sum_{n=1}^\infty \frac{(-1)^n}{2^n-1} & = - \sum_{n=1}^\infty \frac{(-1)^n2^{-n}}{1-2^{-n}} \\
& = - \sum_{n=1}^\infty (-1)^n\sum_{k=1}^\infty (2^{-n})^k \\
& = \sum_{k,n = 1}^\infty (-1)^{n+1} 2^{-nk}.
\end{align}

Similarly, for the right hand side we use $$\frac{q}{1+q} = \sum_{k=1}^\infty (-1)^{k+1} q^k $$ to get
\begin{align}
\sum_{n=1}^\infty \frac{1}{1+2^n} &= \sum_{n=1}^\infty \frac{2^{-n}}{1+2^{-n}}\\
& = \sum_{n=1}^\infty \sum_{k=1}^\infty (-1)^{k+1}(2^{-n})^k \\
& = \sum_{k,n=1}^\infty (-1)^{k+1} 2^{-nk}.
\end{align}