To ensure that headfont matches the font settings of chapter and section titles in a KOMA-Script document use
\usekomafont{disposition}
The initial definition for the disposition font element is \normalcolor\sffamily\bfseries. But this can be changed using \setkomafont or \addtokomafont.
\documentclass{scrbook}
\PreventPackageFromLoading{remreset}% comment this line, if you use an older TeX-Distribution
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amsthm}
%\addtokomafont{disposition}{\rmfamily}
\usepackage{thmtools}
\declaretheoremstyle[headfont=\usekomafont{disposition}]{thm}
\declaretheorem[style=thm]{thm}
\begin{document}
\section{Introduction to the theorem}
Introduction of something. Here's the theorem:
\begin{thm}[Note]
This is a theorem.
\end{thm}
\end{document}
Additional remark: thmtools loads package remreset which is obsolete with uptodate TeX distributions. To avoid the resulting warning »The remreset package is obsolete: \@removefromreset is defined.« KOMA-Script command \PreventPackageFromLoading{remreset} is used in the example.
bodyfont=\itshape? – Richard Costa Jun 10 '19 at 17:53