How do I add a line break in display math mode?

The \[...\] is used to typeset a single equation, not multiple equations.

You need to use an environment that allows for multiple equations. One way is to use gather, but I normally use align, both from the amsmath package:

Notes:

• With display math environments if you leave blank lines in between you get additional vertical spacing and the possibility of a page break between a paragraph and following display.
• If you don't want the equations numbered use the starred versions of the environments: gather* and align*.
• A really good reference is Herbert Voss's comprehensive review of mathematics in (La)TeX.

Code:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{gather}
    a + b = c \\ 
    a = c - b
\end{gather}
%
With the align environment you can align equations:
%
\begin{align}
    a + b &= c \\        
        a &= c - b
\end{align}
\end{document}

In case gather is not an option, for example, an imported command where everything is in math mode, consider make them a single-columned matrix without showing the symbol, then you can use \\.
Code:

\[
\begin{array}{c}
a + b = c \\  % just use \\
a = c - b
\end{array}
\]