Hartshorne Proposition III.9.8, understanding associated points and extensions From Hartshorne III
Prop 9.8 Let Y be a regular scheme of dimension 1, let P $\in Y$ be a closed point, and let X $\subset \mathbf P_{Y-P}^n$ be a closed subscheme which is flat over Y - P. Then there exists a unique closed subscheme $\bar X \subset \mathbf P_Y^n$, flat over Y, whose restriction to $\mathbf P_{Y-P}^n$ is X.
The proof consists of taking $\bar X$ to be the set-theoretic closure of X. My trouble is in understanding why the associated points of $\bar X$ are the same as the associated points of X, and why any other extension would have some associated points mapping to P. How can other extensions look like?
By the way, how should I actually think about associated points? The only visual example that I have in mind is the "cross". Let A := k[x,y]/(xy), then the points (x), (y) and (x,y), only the last of which is closed, are all associated points because every element in the corresponding prime ideals is a zero-divisor. Does this mean that I should think of associated points as points that have some intersection with "another irreducible component"?

In the ring $A=k[X,Y]/(X\cdot Y)=k[x,y]$  the prime ideals $(x),(y)$ are indeed associated, since they are minimal.
The  maximal ideal $(x,y)$ however is not associated since in a reduced  noetherian ring the converse of the above  is true: any associated prime is a minimal prime.    
Ravi Vakil in his magnificent algebraic geometry course   has a wonderful explanation of why and how the associated points of a scheme are its most important points.
Since it would be grotesque for me to try to improve upon his explanations, I will just direct you to page 164 of these notes.
