Hello.
New to the site, so be gentle.
I can think of two reasons driving the deprotonated FMOC forming the double bond and ejecting the protected peptide: greater stability of the product and peptide as a good leaving group.
First is the deprotonated FMOC - it has 14 pi electron, making the entire molecule aromatic (Hückels rule: 4 x 3 + 2 = 12 + 2 = 14 --> one aromatic system). However, the the pi-electrons of an external double bond do not add electrons to the aromatic system. Twelve electrons do not make an aromatic system (4n+2 = 12 --> n = 5/2), so the diphenyl-part are seperate. Now the molecule has two aromatic groups instead of one. Ie. formation of the double bond makes the molecule more stable.
The second reason is that protected peptide is a good leaving group, as macman104 explained before. Also, the negative charge is delocalized among the COO
--system.
Hopefully I cleared something up