Why is the diffeomorphism group a manifold?

Question

Let $M$ be a differentiable manifold. The diffeomorphism group of $M$ is the group of all $C^{\infty}$ diffeomorphisms of $M$ to itself, denoted by $\text{Diff}(M)$. This space of diffeomorphism $\text{Diff}(M)$ is considered as a Lie Group equipped with composition of functions ($\circ$) as a group operation. According to the definition of Lie Group, this implies that $\text{Diff}(M)$ is a manifold. I was wondering what is the intuition or proof that shows that $\text{Diff}(M)$ is indeed a manifold?