Apparently, this is based on a result by Iwahori and Matsumoto (Corollary 2.17 of [IM]). A modern proof of this result can be found on [DHLH]. This result is one of the main elements of the proof of the Hilbert-Mumford criterion (see page 52 of [GIT]).

What the theorem of Iwahori and Matsumoto asserts is that any element $g\in LG$ can be written as $g=h_1 t^\mu h_2$, for $h_1, h_2 \in L^+G$, $\mu: \mathbb{G}_m\rightarrow G$ a 1-parameter subgroup and $t^\mu$ the image of $t\in \mathbb{C}((t))$ by the associated map $\mu(\mathbb{C}((t))): \mathbb{C}((t))^\times \rightarrow LG$.

Since maximal tori are conjugate, the orbit $L^+G t^\mu L^+G$ associated to any 1-PS can be labeled by a coweight $X_*(T)$. Moreover, we can further identify two orbits if the coweights are conjugate, so we can quotient by the Weyl group and get $X_*(T)/W=X_*(T)^+$.

A good reference for this topic, where I originally found about the paper of Iwahori-Matsumoto is section 4.5 of the book [BD].

I still do not know if there's an easier way of proving this (i.e. without using the theorem of Iwahori-Matsumoto) and do not get in what way this is a Cartan decomposition. Anyway, this solves my question.

**References:**

[AHLH] Alper, J., Halpern-Leistner, D., & Heinloth, J. (2019). Cartan-Iwahori-Matsumoto decompositions for reductive groups. arXiv preprint arXiv:1903.00128.

[BD] Beilinson, A.; Drinfeld, V. Quantization of Hitchin's integrable system and Hecke eigensheaves. https://math.uchicago.edu/~drinfeld/langlands/QuantizationHitchin.pdf

[IM] Iwahori, Nagayoshi; Matsumoto, Hideya. On some Bruhat decomposition and the structure of the Hecke rings of $p$-adic Chevalley groups. Publications Mathématiques de l'IHÉS, Tome 25 (1965) , pp. 5-48. http://www.numdam.org/item/PMIHES_1965__25__5_0/

[GIT] Mumford, D. ; Fogarty, J. ; Kirwan, F. Geometric invariant theory.
Third edition.