What is the point of taking hostages in Payday?

The hostages system gives you the opportunity to free one of your teamates from jail. Since you can't really "die" in Payday, this is their "revival" system.

You can take civilians as hostages by shouting at them and zip tying them, or even cops. Taking cops as hostages is a bit more tricky; you may need to hit them first (with a non-lethal shoot or with your melee attack) and then shout at them until they handcuff themselves, which will save you a pair of zip ties. Also note that you cannot accomplish that during an assault (indicated by the red triangle at the bottom of the screen).

If one of your teammates gets arrested, you'll have to wait to the end of the assault plus the eventual time malus (if they killed civilians) to proceed to the trade.

Be careful with your hostages, as they are slow to be moved. So if you leave them without surveillance, they can be freed by the cops.

If one of the players gets arrested (aka killed), Bain (the guy who gives your orders) will start negotiating with the police.

After a while (depending on how many civilians were killed), a hostage is selected and if you free him/her, the arrested player respawns.