Although the notation and explanation of the paper is terrible, it's not supposed to be a very deep fact.

Indeed, let's assume for simplicity that $T = \mathbb G_m^n$ is a *split* torus. Then arbitrary morphisms of schemes $\mathbb G_m \to T$ correspond to $n$-tuples $(f_1,\ldots,f_n) \in (k[x^{\pm 1}]^\times)^n$. Under this correspondence, the characters correspond to the monomials
$$\{(x^{a_1},\ldots,x^{a_n})\ |\ a_1,\ldots,a_n \in \mathbb Z \}.$$
What the authors mean by *formal loops* $\phi \colon \mathbb G_m \to T$ is morphisms $\operatorname{Spec} k((x)) \to T$, where $\operatorname{Spec} k((x))$ can be thought of as $\operatorname{Spec}\left(k[x^{\pm 1}] \otimes_{k[x]} k[[x]]\right)$ (i.e. $\mathbb G_m \times_{\mathbb A^1} \hat{\mathbb A}^1$; the 'formal completion of $\mathbb G_m$ at the origin'). This is an algebraic geometry version of looking at very small loops.

Algebraically, the elements of $L(T)$ are given by $n$-tuples $(f_1,\ldots,f_n) \in (k((x))^\times)^n$. The condition on $L_0(T)$ is that the $f_i$ are invertible elements of the power series ring $k[[x]]$.

Finally, the claim they make just boils down to the statement that any $(f_1,\ldots,f_n) \in (k((x))^\times)^n$ can be written uniquely as a product
$$(f_1,\ldots,f_n) = (g_1,\ldots,g_n) \times (h_1,\ldots,h_n),$$
where the $g_i$ are monomials (coming from $X_*(T)$) and the $h_i$ are invertible elements of $k[[x]]$ (coming from $L_0(T)$). Explicitly, if
$$f_i = \sum_{j\in\mathbb Z} a_j x^j,$$
then $g_i = x^{j_0}$ for the minimal $j_0$ such that $a_{j_0} \neq 0$, and
$$h_i = \frac{f_i}{g_i} = \sum_{j = 0}^\infty a_{j+j_0} x^j$$
is invertible in $k[[x]]$ since its constant term $a_{j_0}$ is nonzero.