Why does SYBR green emit more signal when bound to DNA?
In dyes such as this rotation about bonds that can be involved in the conjugated part of the molecule contribute to radiationless transitions causing the excited state to transfer its energy to  a triplet or back to the ground state rather than fluorescing. When intercalated between base pairs as you suggest in your diagram, the molecule is held more rigidly and so radiationless transfer is reduced and fluorescence increased. 

In molecules with a dipole it may additionally be the case that in a polar solvent the stabilisation due to the solvent polarity brings the initially formed excited state closer to a crossing to a charge-transfer state and this non-radiative process will lower fluorescence yield. Intercalated in the DNA the polarity will be lower than in water and again this will result in more fluorescence.  If your molecule changes wavelength in polar vs non-polar solvents then this is indicative of this process.