The reason blue tits are socially monogamous is that it takes two to successfully raise a brood of chicks.

Females need to be fed when they are about to lay as they cannot fly very easily and it makes it harder for them to find food. They also can't leave the eggs, or they'll get cold and not hatch. Once laid the chicks need to be fed by both parents just to survive.

There are strong benefits males can offer females. Hard-working males, who put in lots of effort to find and deliver food to mum and chicks, means a better chance of survival. Males with good territories will find this easier to do than males in less good territories, and healthy males will be able to do this with less effort than unhealthy males whatever the territory is like.

From the female point of view the most important thing is how much food she gets.

The male can prove his effort and ability by courtship feeding (offering her food during wooing - before they even set up home together), by having a nice territory with lots of food and by singing energy demanding songs (proving he's healthy).