To be able to use \left[ and \right] inside an equation with linebreaks you have to use package breqn:
\documentclass{article}
\usepackage{amsmath}
\usepackage{slashed}
\usepackage{breqn}
\newcommand*{\tr}{\ensuremath{\mathrm{tr}}}
\begin{document}
\begin{dmath}
R^{1}_{\mu\nu} = \int \frac{d^4p}{(2\pi)^4} \tr \left[ \frac{1}{\slashed{p} - \slashed{k}_2-m} \gamma_{5} \gamma_{\nu} \frac{1}{\slashed{p} - \slashed{q}-m} \gamma_{\mu} \\
- \frac{1}{\slashed{p} -m} \gamma_{5} \gamma_{\nu} \frac{1}{\slashed{p} - \slashed{k}_1-m} \gamma_{\mu} \right]
\end{dmath}
\end{document}

An alternative would be, to use either \right. and \left., but with the risk of having different size:
\documentclass{article}
\usepackage{amsmath}
\usepackage{slashed}
\newcommand*{\tr}{\ensuremath{\mathrm{tr}}}
\begin{document}
\begin{equation}
\begin{aligned}
R^{1}_{\mu\nu} = \int & \frac{d^4p}{(2\pi)^4} \tr \left[
\frac{1}{\slashed{p} - \slashed{k}_2-m}
\gamma_{5} \gamma_{\nu} \frac{1}{\slashed{p} -
\slashed{q}-m} \gamma_{\mu} \right. \\
& \left. - \frac{1}{\slashed{p} -m} \gamma_{5} \gamma_{\nu} \frac{1}{\slashed{p} - \slashed{k}_1-m} \gamma_{\mu} \right]
\end{aligned}
\end{equation}
\end{document}

or to explicitly use a size:
\documentclass{article}
\usepackage{amsmath}
\usepackage{slashed}
\newcommand*{\tr}{\ensuremath{\mathrm{tr}}}
\begin{document}
\begin{equation}
\begin{aligned}
R^{1}_{\mu\nu} = \int & \frac{d^4p}{(2\pi)^4} \tr \biggl[
\frac{1}{\slashed{p} - \slashed{k}_2-m}
\gamma_{5} \gamma_{\nu} \frac{1}{\slashed{p} -
\slashed{q}-m} \gamma_{\mu} \\
& - \frac{1}{\slashed{p} -m} \gamma_{5} \gamma_{\nu} \frac{1}{\slashed{p} - \slashed{k}_1-m} \gamma_{\mu} \biggr]
\end{aligned}
\end{equation}
\end{document}

\left[and\right]doesn't work over two (or more) lines. In such cases you should use\bigl(orBigl˛,biggl`, etc) instead. – Zarko Jan 30 '23 at 08:04\left(...\right), make sure you are aware of the issues: here is a small list of issues, here's a discussion about spacing, and here's a demonstration why \left(\right) is problematic. – Celdor Jan 30 '23 at 11:24