Shield Bestowal is a passive upgrade to the already potent Advent Hangar Defense, unlockable through research. This allows the Advent Hangar Defense to passively protect nearby structures with shields and said shields regenerating 4 points a second. This bonus does not stack with other hangar's Shield Bestowal ability.
A player might think that this upgrade would be of little significance, but an extra four hitpoints a second are very significant in long engagements. In short, brutal attacks make this upgrade rather trivial, considering that most structures do not last long once the full attention of the fleet is on them even with shields. But for countering AOE weapons, this can be very critical. For example, the Advent's Destra Crusader's Ruthlessness passive ability is completely canceled out by this passive ability of the Hangar Defense. However, the biggest benefit Shield Bestowal provides is the addition of Shield Mitigation to nearby structures, that would be shieldless otherwise. This effectively doubles their durability.
It's worth noting that Strike Craft have no shields, and therefore are unaffected by this ability. In order to patch a mangled Anima, you have to dock it until it's fixed. This bonus only affects stationary objects.