A new line inside subscript

You can use \substack

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\[ A_k = \bigcup_{\substack{I \subset \{ 1,\dots,k \}\\ \text{card} J = K}} B_j\]
\end{document}

If you feel that there is lot of empty space, here is how to get it out. This one uses \mathclap from mathtools.

\documentclass{article}
\usepackage{mathtools}

\begin{document}
\[ A_k = \bigcup_{\mathclap{\substack{I \subset \{ 1,\dots,k \}\\ \text{card} J = K}}} B_j\]
\end{document}

Response to comment

It works inside align*:

\documentclass{article}
\usepackage{mathtools}

\begin{document}
\begin{align*}
    A_k &= \bigcup_{\mathclap{\substack{I \subset \{ 1,\dots,k \}\\ \text{card} J = K}}} B_j\\
    A_k &= \bigcup_{\mathclap{\substack{I \subset \{ 1,\dots,k \}\\ \text{card} J = K}}} B_j
\end{align*}
\end{document}

While not as streamlined as the accepted solution, I add this solution to the mix, because it shows the 2-line subscript left justified, as was provided in your example, which the accepted solution (to this point) does not. I also made the subscript in \scriptscriptstyle, which may be preferable for a multi-line subscript. Vertical gaps between the stacked lines are adjustable.

\documentclass{article}
\usepackage{stackengine}
\usepackage{amsmath}
\parskip 1em
\begin{document}
\[ \def\useanchorwidth{T}\stackMath
  A_k = \mathop{\stackunder{\bigcup}{\def\stackalignment{l}%
    \stackunder[2pt]{\scriptscriptstyle I \subset \{ 1,...,k \}}
                    {\scriptscriptstyle\text{card} J = K}
  }} B_j\]

\[ \def\useanchorwidth{F}\stackMath
  A_k = \mathop{\stackunder{\bigcup}{\def\stackalignment{l}%
    \stackunder[2pt]{\scriptscriptstyle I \subset \{ 1,...,k \}}
                    {\scriptscriptstyle\text{card} J = K}
  }} B_j\]
\end{document}