Can someone what this notation means? 


I don't understand what does $\phi_I$ mean
The proof includes writing $\phi_I$ as a product of $\phi_{i_1}\phi_{i_2},\dots$, but it doesn't explain what the LHS really means
 A: $\phi_I$ is a $k$-multilinear form. This means $\phi_I$ takes in a $k$-tuple of vectors and is linear in slot if you fix the others. For example: 
$$\phi_I(c{\bf v}+{\bf w},{\bf v}_2,\dots,{\bf v}_k) = c\phi_I({\bf v},{\bf v}_2,\dots,{\bf v}_k)+ \phi_I({\bf w},{\bf v}_2,\dots,{\bf v}_k)$$
where $c$ is a scalar and the v's are vectors. Likewise, $\phi_I$ is linear in the other arguments. By "form" I mean that the output: $\phi_I({\bf v}_1,\dots,{\bf v}_n)$ is a scalar ($\phi_I$ maps $k$-tuples of vectors to scalars).
Now $I$ is a $k$-tuple of integers. Given a list of vectors from the given basis: $a_{j_1},\dots,a_{j_k}$ (repeats allowed), $\phi_I(a_{j_1},\dots,a_{j_k})$ spits out 0 unless $(j_1,\dots,j_k)=I$ (the subscripts match identically).
For example: $\mathbb{R}^2$ has the basis ${\bf i}=(1,0)$ and ${\bf j}=(0,1)$.
Then...
$$\phi_{(1,1)}({\bf i},{\bf i})=1, \quad \phi_{(1,1)}({\bf i},{\bf j})=0, \quad \phi_{(1,1)}({\bf j},{\bf i})=0, \quad \phi_{(1,1)}({\bf j},{\bf j})=0$$
$$\phi_{(1,2)}({\bf i},{\bf i})=0, \quad \phi_{(1,2)}({\bf i},{\bf j})=1, \quad \phi_{(1,2)}({\bf j},{\bf i})=0, \quad \phi_{(1,2)}({\bf j},{\bf j})=0$$
$$\phi_{(2,1)}({\bf i},{\bf i})=0, \quad \phi_{(2,1)}({\bf i},{\bf j})=0, \quad \phi_{(2,1)}({\bf j},{\bf i})=1, \quad \phi_{(2,1)}({\bf j},{\bf j})=0$$
$$\phi_{(2,2)}({\bf i},{\bf i})=0, \quad \phi_{(2,2)}({\bf i},{\bf j})=0, \quad \phi_{(2,2)}({\bf j},{\bf i})=0, \quad \phi_{(2,2)}({\bf j},{\bf j})=1$$
Also, for example, 
$$\phi_{(1,2)}((1,2),(3,4)) = \phi_{(1,2)}({\bf i}+2{\bf j},3{\bf i}+4{\bf j})
=\phi_{(1,2)}({\bf i},3{\bf i}+4{\bf j})+2\phi_{(1,2)}({\bf j},3{\bf i}+4{\bf j})$$
$$=3\phi_{(1,2)}({\bf i},{\bf i})+4\phi_{(1,2)}({\bf i},{\bf j})+6\phi_{(1,2)}({\bf j},{\bf i})+8\phi_{(1,2)}({\bf j},{\bf j})=3(0)+4(1)+6(0)+8(0)=4$$
