Intuition behind the definition of the Siegel-Eichler transformation
This is explained very clearly in Wall's paper, Diffeomorphisms of 4-manifolds, J. London Math. Soc. (1964) 131-140. The idea is that if you do surgery on a circle $C$ in a simply-connected $4$-manifold $X$, you get $X \# T_r$ where $T$ is a 2-sphere bundle over $S^2$ with fiber $S^2$, and intersection form generated by $x,y$ as above, except that maybe $y\cdot y = r$ may be non-zero. (Up to diffeomorphism, only the parity of $r$ matters.) Now if you perform an isotopy of that $C$, ending up at $C$, you get a self-diffeomorphism of $X \# T_r$. (You can get $r\neq 0$ by twisting the framing of the circle in this isotopy.) If the $2$-dimensional homology class swept out by the circle (ie a map of a torus) in this isotopy is $\omega$, then the induced map on homology is of the form $E^1_\omega$.
I don't know what was the original intuition behind the algebraic formula (due to Siegel), but Wall's geometric picture makes it a lot clearer to me.