Prior to the rise of the Shogunates, courtiers after power (more power than the emperor) did so by having the emperor marry one of his daughters. This would result in the next emperor being his grandson, who would naturally defer to his mother and grandfather, often without complaint. It didn't work forever for the power-hungry family, but due to this mechanism, said family could rule Japan in the name of the Emperor, using his legitimacy as the descendant of the sun goddess while simultaneously monopolizing the imperial regent position and appointing family members to important positions. Said family would be able to rule Japan in practice, without having to risk a violent overthrow of the emperor. This of course also had the effect of preserving the imperial line.

For after the rise of the Shogunates, when this marriage-rule stopped (or rather, it continued, but stopped being useful for controlling the country) please see this thread

TL;DR: In no particular order

1. By the time of the Shogunates, the Japanese were already very much used to having the Emperor not having too much personal power and having others rule in his name. Also, not touched on in the thread, the samurai were used to working for the court.
2. The Emperor (and court) had strong symbolic value and legitimacy that the Shogunates knew and may even have also believed in.
3. Neither the Kamakura nor the Muromachi Shogunates had the power to remove the Emperor without probably throwing the country into chaos from which they might not end up winning. The Edo Shogunate might have been strong enough to do so eventually, but by then it was more pain than it was worth.