Unwanted separation in an equation

mathtools provides the macro \mathclap (and also \crampedclap, \smashoperator) for precisely this occasion (see pages 4–7 of the manual):

\usepackage{mathtools}
%...
\begin{equation*}
  \langle H, H'\rangle = \bigcap_{\mathclap{\substack{
  L \leq G, \\
  H \cup H' \subseteq L.
  }}}L
\end{equation*}

Try \usepackage{mathtools} and

\[
\langle H, H'\rangle = \smashoperator{\bigcap_{\substack{
L \leq G, \\
H \cup H' \subseteq L.
}}}L
\]