At page 9 of Barrett's elementary differential Geometry he says
Such that $V=v_1U_1+v_2U_2+v_3U_3$
Proof: 
How does he go from the first equation to the second .He didnt prove there is a basis for all vector fields $V$ at $p$ .
All i can understand is that the $$(v_1(p),v_2(p),v_3(p))_p=(v_1(p),v_2(p),v_3(p))+p$$ how do i continue? to get where he got?
In the book he defines a basis at $p$, namely just the one you get from attaching the unit basis vectors $U^i$ to the point $p$. If $V$ is a vector field then (for your purposes) $V: U \to T_p \mathbb{R}^n$ defined by $p \mapsto V_p$. Since $V_p$ is a tangent vector at $p$ and $\{U^1(p),...,U^n(p)\}$ form a basis then we can write;
$$\\$$
$$V_p = (v^1(p),...,v^n(p)) = (v^1,...,v^n)_p = (v^1(1,0,...,0)+ \cdots + v^n(0,.....,1))_p$$
$$ \hspace{2.8in} = v^1(1,0,....,0)_p+ \cdots + v^n(0,.....,1)_p$$
$$\hspace{1.7in} = v^1U^1(p) + \cdots + v^nU^n(p)$$
You should really think about what is being said geometrically (this whole, creating a frame at $p$) then the computations will make sense.