my question is the following:
Let $M$ be a connected differentiable manifold of dimension n, and let $X_1,...,X_n$ be n vector fields which are independent at every point of $M$ and satisfy $[X_j, X_k]=0$ on $M$ for $j,k=1,...,n$, where $[X_j, X_k]$ denotes the commutatore between the vector field.
Is it true that these vector fields are tangent to $M$? Can anyone give me an intuition explanation of this fact?
I believe that you want to say that if $X_1,...,X_k$ are vectors and commutes they are tangent to a SUBMANIFOLD. To see this consider the flow $\phi_{X_i}$ of $X_i$, and the map $\phi:\mathbb{R}^k\rightarrow M$ defined by $\phi(t_1,...,t_k)=\phi_{X_1}(t_1)\circ...\circ\phi_{X_k}(t_k)$ the image of the neighborhood of $0$ is a submanifold.
This is a step in the proof of the Frobenius theorem which asserts if $X_1,...,X_k$ is involutive, then it defines a foliation. To prove this theorem, one can prove the existence of $Y_1,...,Y_k$ such that $Vect(X_1(x),...,X_k(x))=Vect(Y_1(x),...,Y_k(x))$ and $[Y_i,Y_j]=0$.