It seems like you already can see this, but Ext^1(M,N) is measuring all the ways to form distinct short exact sequences 0 -> N -> ? -> M -> 0

If you are looking for intuition, what you should do is think about picking a set of generators G for M. This is equivalent to choosing a surjection from the free module R^G -> M, whose kernel is the 1st syzygy module S (which you should think of as relations between the generators). Then Ext^1(M,N) is going to be all the maps from S to N, modulo those which come from a map R^G to N. Therefore, Ext^1 is something like all the ways of assigning an element in N to every relation between generators in M, modulo dumb ways of doing this assignment (those coming from sending generators to N). This perspective can make it easy to guess what Ext^1 should be.

It seems like you already can see this, but $Ext^1(M,N)$ is measuring all the ways to form distinct short exact sequences $0\to N\to ?\to M\to 0$

If you are looking for intuition, what you should do is think about picking a set of generators $G$ for $M$. This is equivalent to choosing a surjection from the free module $R^G \to M$, whose kernel is the 1st syzygy module S (which you should think of as relations between the generators). Then $Ext^1(M,N)$ is going to be all the maps from $S$ to $N$, modulo those which come from a map $R^G \to N$. Therefore, $Ext^1$ is something like all the ways of assigning an element in $N$ to every relation between generators in $M$, modulo dumb ways of doing this assignment (those coming from sending generators to $N$). This perspective can make it easy to guess what $Ext^1$ should be.

It seems like you already can see this, but Ext^1(M,N) is measuring all the ways to form distinct short exact sequences (your diagram is backwards) 0 -> N -> ? -> M -> 0

If you are looking for intuition, what you should do is think about picking a set of generators G for M. This is equivalent to choosing a surjection from the free module R^G -> M, whose kernel is the 1st syzygy module S (which you should think of as relations between the generators). Then Ext^1(M,N) is going to be all the maps from S to N, modulo those which come from a map R^G to N. Therefore, Ext^1 is something like all the ways of assigning an element in N to every relation between generators in M, modulo dumb ways of doing this assignment (those coming from sending generators to N). This perspective can make it easy to guess what Ext^1 should be.

It seems like you already can see this, but Ext^1(M,N) is measuring all the ways to form distinct short exact sequences 0 -> N -> ? -> M -> 0

If you are looking for intuition, what you should do is think about picking a set of generators G for M. This is equivalent to choosing a surjection from the free module R^G -> M, whose kernel is the 1st syzygy module S (which you should think of as relations between the generators). Then Ext^1(M,N) is going to be all the maps from S to N, modulo those which come from a map R^G to N. Therefore, Ext^1 is something like all the ways of assigning an element in N to every relation between generators in M, modulo dumb ways of doing this assignment (those coming from sending generators to N). This perspective can make it easy to guess what Ext^1 should be.