The following is on Page 4 of Representation Theory A Combinatorial Viewpoint by AMRITANSHU PRASAD:
I am confused by this equation: if $f$ is an element in $K[G]$ (the group algebra), and $g$ is an element in the group $G$, then what does $f(g)$ mean? How about $f(g)\rho(g)$ then, where $\rho(g)$ is an element in $\mathrm{GL}(V)$?
an element $f$ of the group algebra $K[G]$ is just a $K$-linear combination of elements of $G$. The notation $f(g)$ is clumsy. write $f_g$ instead and then you have (with each $f_g \in K$): $$ f = \sum_{g \in G} f_g g $$
thus a representation, which is a group homomorphism, can be extended (by $K$-linearity) to a $K$-algebra homomorphism