We defined the tensor product of $v$ and $w$ to just mean a "symbol" $e_{vw}$, then considered the subspace spanned by all these symbols, and finally we quotient out by relations to make the tensor product between two vectors bilinear. The resulting vector space is the tensor product of $V$ and $W$.
I don't understand why this is a vector space.. sure, we've said it's the span of a bunch of symbols, but what does it even mean to "add" together two symbols? How is equality defined in our vector space?
If $v_1,\cdots v_n$ is a basis of $V$ and $w_1,\ldots w_m$ is a basis of $W$, then $$ \{v_i\otimes w_j\mid 1\le i\le n,1\le j\le m\} $$ is a basis of $V\otimes W$. So we write $e_{vw}=e_v\otimes e_w$.