Lost Bags are mainly a reward from Tokyo Death Metro (TDM), and can contain a variety of things, like KC, SPL, Golden Beasts, R&D Materials, and exclusive TDM equipment and decals.
Higher rarity bags have rewards that lower ones do not. Some Blueprints and Decals may only be obtained from Lost Bags. Every 3 months, there is a new season of TDM Battle Rush with new gear that may only be found in Lost Bags.
Sources
You'll earn up to 5 Lost Bags at the end of each week (Monday 03:00 PDT) based on your rank in Tokyo Death Metro. Your rank will automatically decrease a bit at the start of each week, so make sure to rank up again if you want those Lost Bag rewards!
Brown, Silver and Gold Lost Bags may be obtained from normal TDM Attacks and Defenses. They can also be rewarded after finishing extraction on a captured Fighter. Brown bags are more common rewards from lower TDM Ranks, while Diamond Rank will have the best chances of rewarding you with Silver or Gold bags.
Gold and Rainbow Lost Bags are also rewards from Weekly and Season Quests. If you did all of them, that would be 30 Gold Bags and 3 Rainbow Bags over the course of 3 months, 2 Gold Bags each week, as well as the TDM Rank Lost Bags each week.
Platinum Lost Bags may only be obtained as a TDM Rank Reward from Platinum I or Diamond III / II / I. Some gear may only be found in Platinum Bags, so the best way to obtain them is to reach Diamond I for 4 Platinum Bags each week.
Rainbow Bags are only obtained from the Season Quests, or from Store purchases that include them. Other Bags may appear in the Store, but Rainbows are exclusive in this way.
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