The package exercise has two \expandafter commands in the wrong place; they're actually wrong and must be removed.
There's a telltale warning you get:
LaTeX Font Warning: Font shape `T1/jkp/m/i' undefined
(Font) using `T1/jkp/m/n' instead on input line 1.
which stems from the line in the loe file
\contentsline {exercise}{{\"U}bung\ 1\ {\relax \fontshape it\selectfont Integration 1}\hspace {.66em}}{1}{}%
Obviously \fontshape it is wrong and is produced by the wrong \expandafter we'll remove.
\documentclass[11pt, oneside]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{makeidx}
\usepackage{graphicx}
\usepackage{lmodern}
\usepackage{kpfonts}
\usepackage[]{geometry}
\usepackage{lipsum}
\usepackage{xcolor}
\usepackage{framed}
\usepackage{exercise}
\usepackage{etoolbox}
\makeatletter
% remove the faulty \expandafter
\patchcmd{@@@ExeEnv}
{\theExercise\ \expandafter}
{\theExercise\ }
{}{}
\patchcmd{@@@ExeCmd}
{\theExercise\ \expandafter}
{\theExercise\ }
{}{}
\makeatother
\begin{document}
\tableofcontents
\listofexercises
\section{Lipsum}
\lipsum[1]
\begin{framed}
\begin{Exercise}[
title={Integration 1},
difficulty=0,
origin=myself, label=aufg1]
Integrieren Sie folgende Funktion:
[\left(\sin (x)+\cos(x)\right)^{e^{-x}}]
\end{Exercise}
\end{framed}
\lipsum[1]
\end{document}
I added \usepackage[T1]{fontenc} that's really necessary for typesetting in German.
Now the loe file will have the right incantation:
\contentsline {exercise}{{\"U}bung\ 1\ {\itshape Integration 1}\hspace {.66em}}{1}{}%

If you also want to remove \itshape, do a second series of patches.
\documentclass[11pt, oneside]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{makeidx}
\usepackage{graphicx}
\usepackage{lmodern}
\usepackage{kpfonts}
\usepackage[]{geometry}
\usepackage{lipsum}
\usepackage{xcolor}
\usepackage{framed}
\usepackage{exercise}
\usepackage{etoolbox}
\makeatletter
% remove the faulty \expandafter
\patchcmd{@@@ExeEnv}
{\theExercise\ \expandafter}
{\theExercise\ }
{}{}
\patchcmd{@@@ExeCmd}
{\theExercise\ \expandafter}
{\theExercise\ }
{}{}
% remove \itshape
\patchcmd{@@@ExeEnv}
{\itshape}
{}
{}{}
\patchcmd{@@@ExeCmd}
{\itshape}
{}
{}{}
\makeatother
\begin{document}
\tableofcontents
\listofexercises
\section{Lipsum}
\lipsum[1]
\begin{framed}
\begin{Exercise}[
title={Integration 1},
difficulty=0,
origin=myself, label=aufg1]
Integrieren Sie folgende Funktion:
[\left(\sin (x)+\cos(x)\right)^{e^{-x}}]
\end{Exercise}
\end{framed}
\lipsum[1]
\end{document}

\expandafter. Now I realize I didn't change the second one! Oops! – egreg Jul 30 '21 at 21:51