One perspective to see why the Hahn-Banach theorem is 

considered as the first big theorem in linear functional analysis. 

The geometric version of the Hahn-Banach theorem is the 

hyperplane separation theorem (geometrically, the theorem 

is so obvious as if it says nothing). 

Lots of fundamental theorems 

in many applied fields (optimization, finance, economics, game 

theory), the key step is always based on the hyperplane 

separation theorem. 

Logically, it is not difficult to follow the proofs, 

but it is not easy to develop a high-level understanding of those 

proofs (I mean, not easy to see why the key step always eventually 

relies solely on a theorem that seems to say nothing).

Comment: It seems to say nothing because you are thinking 

in ordinary 3-dimensional geometry, for which the result is trivial.  

But when the space is a function space the result is much more 

meaningful as one can define spaces for which it does not hold.  

And actually, there are even relatively simple spaces for which it 

fails.  The point is that it is a sort of minimal property for which 

"nice" things happen.