Let $k$ be a field with $char(k)\neq p$. Take a representation of the absolute Galois group $G=Gal(k_{alg}/k)$ and a continuous representation $V$ of $G$, with coefficients in, say, $\mathbb{Z}_p$.
Now if $\mathbb{Z}_p(1)$ the Tate module of $p$th-power roots of unity in $k_{alg}$, one usually denotes by $V(1)$ the representation $V\otimes_{\mathbb{Z}_p}\mathbb{Z}_p(1)$ and set $V(-1)=Hom_{\mathbb{Z}_p}(\mathbb{Z}_p(1),V)$.
I get these definitions but i would like some intuition about the reason we consider these twists. What are the main reasons we consider them? What are the main basic applications of these? (i don't mean here deep geometric conjectures but mainly the reasons they were introduced in the first place).