The fundamental notion behind the theory of groups is the concept of symmetries and the best place to visualize symmetries is Euclidean space and, by extension, Hilbert's space.
By a symmetry on a Hilbert space $H$ one means any map
that preserves distance, and sends the origin to itself. In fact any such map is necessarily a unitary operator.
The group $\mathscr U(H)$, formed by all unitary operators on Hilbert's space is, according to this, the archetype of symmetry!
Given any group $G$, it therefore makes a lot of sense to try to model $G$ via $\mathscr U(H)$, and this is usually done by considering
u:G\to \mathscr U(H),
often called group representations.
If $u_1$ and $u_2$ are representations of $G$, one may define a direct sum representation
on $H\oplus H$, but
everything there is to know about $u$ is already present in either $u_1$ or $u_2$.
On the other hand, should we be
interested in studying a given representation $u$, we might ask whether or not it has an expression such as (1), in
which case it
would be better to study $u_1$ and $u_2$ separately.
This means that one should concentrate in the study of irreducible group representations, namely those which cannot
be split as in (1).
For abelian groups, Schur's Lemma says that all irreducible unitary representations are necessarily one-dimensional,
meaning that the underlying Hilbert space is $\mathbb C$, and since
\mathscr U(\mathbb C) = \mathbb T,
we are inexorably led to studying homomorphisms from $G$ to $\mathbb T$!