I just read the fact $M_0(\Gamma)=\mathbb{C}$ (constant functions) where $M_0(\Gamma)$ is the space of modular forms of weight $0$ on the congruence subgroup $\Gamma$.
But in this article at page four under the picture of the fundamental domain, the function $y(t)$ is said to be in $M_0(\Gamma_1(6))$ but $y$ is not constant. Have I misunderstood something?
This is a little confusing. Let $Y_1(6)$ be the open modular curve $\mathcal{H}/\Gamma_1(6)$. It has a canonical compactification $X_1(6)$ on which the vector space of functions is indeed one-dimensional. The field of modular functions is the function field of $X_1(6)$, which has transcendence degree 1 over the complex numbers (and in particular is certainly not 1-dimensional).
Only the constant functions extend holomorphically to all of $X_1(6)$; however, as is stated in the paper, the function $y$ they write down has a unique zero in the fundamental domain (hence a unique pole somewhere, presumably at a cusp although I didn't read the definition of $y$), so gives an isomorphism with $\mathbb{P}^1$.
Or, if you don't like the algebraic language here, the function $y$ has a pole at $1/2$ (as stated in that paper) which corresponds to its $q$-expansion there failing the cuspidal condition of being a modular form.