Why did so many Saxons emigrate to England?

I read some interesting DNA research recently showing that vikings left little or no DNA traces in the population of present day UK but Saxons on the other hand left a huge imprint.

It amazes me that so many could have moved across to England. It would surely involve massive immigration. They must have had large ships. Was Saxony very much overpopulated to have so many people able to move across the sea?