Under the dictionary, subrepresentations are exactly submodules and irreducible representations are exactly simple modules

A linear subspace $W\le V$ is stable under every $g\in G$ if and only if it is stable under every basis element $[g]$ of $k[G]$ under the action of [L2]. Because the action of an arbitrary $a={\sum }_g{r}_g[g]$ is the corresponding $k$-linear combination of the actions of the basis elements, this is equivalent to stability under every element of $k[G]$. So the subrepresentations of $V$ are exactly the $k[G]$-submodules.

By [L1], irreducibility means that $V$ is nonzero and has no proper nonzero subrepresentation. By step 1.1 those are exactly the proper nonzero $k[G]$-submodules, and [L3] is the same condition in module language. Therefore $V$ is irreducible if and only if it is simple as a $k[G]$-module.