From The TimeSplitters Wiki

BagTag is an Arcade mode that features in all three games in the series. This mode is available to play on every map and features just one bag, usually in a central position on the map. The aim is to hold onto the bag for the longest amount of time. Naturally, whoever has the bag instantly becomes the number one target for all other players, and when they are killed they drop the bag for the next person to pick up. The option to return fire whilst carrying the bag can be toggled on or off at the start of the game. Whoever has the longest total time in possession of the bag when the match ends is the winner. In the first TimeSplitters game, you could play BagTag with teams. In this instance, the carrier would accumulate time for their team instead of just themselves.

Tip: If the ability to shoot while carrying the bag is off, you can still throw a grenade or a brick. Equip the brick or grenade and hold fire button (in the case of a brick) or the throw grenade button before grabbing the bag. You can throw it by releasing the button, but once you've thrown it, you can't use another one while holding the bag.