This is Exercise 4.2 in Milne's 'Algebraic Groups':
Let $G$ be an algebraic group over $k$ and $H$ a normal algebraic subgroup of $G$. From a representation $(V,r)$ of $H$ and a $g\in G(k)$, we get a conjugate representation $h \mapsto r(ghg^{-1})$ of $H$. Assume that $G(k)$ is schematically dense in $G$, and let $(V,r)$ be a simple representation of $G$. Show that $(V, r|_H)$ is semisimple and that all of its simple constituents are conjugate and have the same multiplicity.
This is basically a generalization of Clifford's theorem to algebraic groups. The original version says that a simple representation of an abstract group splits as a direct sum of conjugate simple representations after restriction to a normal subgroup. In particular, for any $k-$algebra $R$ it shows the representation $(V\otimes R,r)$ of $G(R)$ splits as a direct sum of conjugate representations of $H(R)$, i.e. representations of the form $W_R^{g_i}$ conjugate to some representation $W_R$, where $W_R^{g_i}$ have the same underlying space as $W_R$ and $h\in H(R)$ acts on $W_R^{g_i}$ as $g_ihg_i^{-1}$ acts on $W_R$ with $g_i\in G(R)$. In particular, this is true for $R=k$.
The only thing that remains is to show that these $g_i$ can be chosen in a consistent way for all $R$, i.e. as base changes of some $g_i\in G(k).$ This is where the schematic density should come into play, but I can't seem to understand how exactly.
I apologise for the messy notation, I still am not quite sure whether spelling everything out explicitly makes the writing more clear or more overburdened with notation when working with algebraic groups as group schemes.