firstly, the set of all vector fields on $G$ is the subalgebra of left invariant vector fields. I read some reference and find that we do research of algebra always by left invariant, my question is why not right invariant? does it make difference?
Secondly, how to understand that they are isomorphic? ($T_eG$ is the tangent space at $e\in G$)
Thank you.
The set of all left invariant vector fields is a subset of the set of vector fields. They are not equal. The reason $T_eG$ is isomorphic to the set of left invariant vector fields is that each vector at the identity extends to a unique left invariant vector field. Each vector field extends to a unique right invariant vector field too, but left is the standard convention.
If you review the definition of left invariant vector field it should be clear how to extend a given vector at the identity to a left invariant vector field. Basically given $v\in T_e G$ we push forward the vector $v$ to the point $g\in G$ using the left multiplication map $L_g :G\to G$ defined by $L_g (h)=gh$. So the extension of $v$ is a vector field $V$ on $G$ defined by $V_g=L_g^* v$ for $g\in G$, where $L_g^*:T_eG\to T_gG$ denotes the pushforward.