Which has the largest bond angle between water, oxygen difluoride and dichlorine oxide?
There is practically only one effect that determines the angle, but there is an underlying mechanism that you also should understand.

Ideally, a central atom such as oxygen would want to use its p-orbitals exclusively for bonding and have the s-orbital electrons sitting comfortably as a lone pair close to its nucleus. However, that requires a $90^\circ$ angle which can create rather strained molecules — remember that each atom is surrounded by an electron cloud with negative charge that repulses other atoms’ electron clouds. There are now two possibilities of how to relieve this steric strain:


enlargen the bond lengths, so that different substituents are further away from each other while keeping a bond angle as perfect as possible;
enlargen the bond angles so the substituents move further apart from each other while retaining a close distance to their central atom with a more perfect bond length.


In practice, almost all systems stabilise themselves by modifying the angles. Increasing bond lengths decreases bond strengths and thus energy gained much more strongly than tinkering around with angles, because the overlap will inevitably get significantly smaller in absolute value number no matter which way you look at it.

An average bond length can be thought of as the distance which is just large enough for those orbitals that interact to do so favourably while minimising nonfavourable interactions. For certain atom pairs, bond length can essentially be considered almost a constant.

All of this taken together means that the very small hydrogen atoms have a much easier time of crowding together closer to the $90^\circ$ angle than the larger fluorine or huge (in this context) chlorine atoms. Even though the latters’ bond lengths are longer in absolute values their larger radii makes the effective bond lengths shorter and thus makes them require more space. Since chlorine is the largest of the lot, the $\ce{OCl2}$ bond angle is largest.