The czech language module for babel makes the hyphen character - an active shorthand, which means that some uses of - break or don't work as expected.
You can temporarily disable this special meaning of - with \shorthandoff, to keep the effect of this command local, we add \begingroup...\endgroup around the equation.
\documentclass{article}
\usepackage{systeme}
\usepackage[czech]{babel}
\begin{document}
\begingroup
\shorthandoff{-}
\[
\systeme{x+2y = 0, x-y = -1}
\]
\endgroup
\end{document}

Since \[...\] comes with a grouping of its own,
\[
\shorthandoff{-}
\systeme{x+2y = 0, x-y = -1}
\]
also works (thanks to @egreg for pointing that out in the comments).
Another option would be to use no grouping and explicitly turn the shorthand back on
\shorthandoff{-}
\[
\systeme{x+2y = 0, x-y = -1}
\]
\shorthandon{-}
Note that in LaTeX \[...\] is preferable over $$...$$, see Why is \[ ... \] preferable to $$ ... $$?.
\shorthandoff{-}before the equation (if you needczech's special meaning of-you should add\shorthandon{-}after the equation or use\begingroup...\endgrouparound the equation to keep the effect local). Note that\[...\]is preferred over$$...$$in LaTeX – moewe Mar 21 '20 at 11:23