Upgraded bridge systems allow a vessel improved chance at escaping from Battles.
The Bridge allows you to escape battles without being destroyed. This is incredibly useful in PvP as you can escape unwinnable battles and still be ready to fight another battle once you escape.
When you escape successfully, you stop fighting with your enemy and you're sent back to your ship screen. Any HP lost, Ammo used, etc. in battle will still be counted when you escape. However, if you fail to escape, your ship immediately blows up and you lose all your health. The Bridge's escape rate is influenced by numerous things. Here is a complete list of what affects your Bridge Escape Rate.
Increases Escape Rate
The Bridge will always have a 100% escape rate if the enemy does not have a Bridge.
Upgrading the Bridge increases your escape rate if the enemy also has a Bridge.
If you send crew with a Pilot stat to the bridge, it'll increase your Escape Rate by how much their Pilot Stat is. So when a crew member with a pilot stat of 12 is sent to a max level bridge, it'll increase the Escape Rate from 60% to 72%.
Decreases Escape Rate
If you don't have a Bridge, your escape rate will always be 0% and instead your escape option is replaced with surrender.
If you don't have any Reactor at maximum HP, your escape rate will always be 0%.