The rules on passive checks (PHB, p. 175) state:
A passive check is a special kind of ability check that doesn’t involve any die rolls. Such a check can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the GM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster.
Here’s how to determine a character’s total for a passive check:
10 + all modifiers that normally apply to the check
Passives are what you as a DM can use to describe who sees (or figures out) what. Instead of actively calling for rolls, which alert the players that there is something significant (or waste everyone's time if you're doing it just to be random) and also allow for failure to get across information you really want/need to convey.
One of the Acquisitions Inc. podcasts included the party showing up at an inn, and as Chris Perkins described the place, he said that Aeofel looked through the curtains and out a window to see a man out in the blizzard closing up a barn or some such thing. There was no roll called for, and while he likely was not using passives for anything in this case it is the sort of thing you'd use it for. The rest of the characters were doing whatever, and you can provide each with a little description of the place, painting a picture for the whole party and giving particularly observant characters a chance to notice something, which their player can then make use of in RP.
Passive investigation can be used the same way to provide hints and clues. Investigation is used actively to deduce clues that the character is considering, just as we'd expect. Passively, it can be the voice in the back of their mind putting things together almost subconsciously, as well as memories springing up upon coming across information. This again works when you want/need to get information to the party without alerting them that "something is here" and also injecting a very real chance of failure. It also keeps things moving along, because players could take forever to figure something out that their character would be far quicker at doing by virtue of being an actual investigative character and hero.