What is the geometrical meaning of a theorem of vector space? We all are familiar with the underlying theorem

Theorem : Let $~\text V~$ be a vector space and $~\text W_1~$ and $~\text W_2~$ be sub-spaces of $~\text V~$. Then
  $$\dim~(\text W_1~+~\text W_2)~=~\dim~(\text W_1)~+~\dim~(\text W_2)~-~\dim~(\text W_1\cap\text W_2)~.$$

Now my question is 

Question : What is the geometrical meaning of the above stated theorem ?

 A: The formula you mention is called Grassmann's formula (curiously I cannot find an English Wikipedia entry on this, thus I linked to the French one). It can be seen as a vector-space version of the inclusion-exclusion principle, which states that, given finite sets $A$ and $B$, 
$$
|A\cup B|= |A| + |B| - |A\cap B|.$$
Here $|\cdot|$ denotes the cardinality. This fits into the general principle that "finite dimensional vector spaces are like finite sets", with "dimension" in place of "cardinality",$^{[1]}$ and gives an interpretation of the formula, in the case of two subspaces. (This is, arguably, a "combinatorial" interpretation, more than a geometrical one).
However, we must be aware that this fails spectacularly for more than two subspaces. The inclusion-exclusion principle for three sets reads 
$$
|A\cup B\cup C| = |A|+|B|+|C| -|A\cap B| - |A\cap C| - |B\cap C| + |A\cap B\cap C|, $$
so it is very natural to conjecture that 
$$
\begin{split}
\dim(W_1+W_2+W_3)&=\dim W_1 + \dim W_2 + \dim W_3 \\ &-\dim(W_1\cap W_2) - \dim(W_1\cap W_3) - \dim(W_2\cap W_3) \\ &+ \dim(W_1\cap W_2\cap W_3)\end{split},$$
which, however, is false. I find this surprising. 

In conclusion, the formula of Grassmann has a geometrical interpretation only in the case of two subspaces.


[1] For example, a manifestation of this principle is the theorem that, if two sets have the same cardinality, then there is a bijection between them; analogously, if two vector spaces have the same dimension, then they are isomorphic. 
A: You are looking at the subspace generated by linear combinations of vectors in $W_1$ and $W_2$. Let's suppose that we are working in a 3D space. The possible subspaces are:


*

*The origin (dimension 0)

*Any line through origin (dimension 1)

*Any plane through origin (dimension 2)

*The entire space (dimension 3)


Now choose any of the two (you can repeat the dimensionality). For example, say two lines. The dimension of each line is 1. If they are not parallel, the dimensionality of the intersection is $0$, otherwise the intersection is the same as any of the lines. So the subspace formed by the parallel lines has dimensionality $1=1+1-1$ (it's the same as any of the lines) or $2=1+1-0$ (they form a plane).
Let's take another example, a line and a plane. If the line is in the plane, the intersection is the line itself, so the subspace has dimension $2=2+1-1$. If the line is not in the plane, the subspace is the entire space $3=2+1-0$.
Two planes can be identical, so $W_1+W_2$ would have dimension $2=2+2-2$. If the planes are not identical, they will intersect along a line. Then they will span the entire space $3=2+2-1$. And so on.
