On page 26 of the DMG, it describes a passive insight check and gives an example of using it in a social encounter. Could you help me with it and how to use it? Step by step would be appreciated
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The passive insight check is not something that is rolled or actively used. It is more like armor class or a reflex defense, a number which the opponent must beat to successfully deceive the character. As such, there is no retrying, just as there is no retrying an armor class or reflex defense.
It is, from a gameplay perspective, a tool to speed up the game and reduce randomness. It's basically the assumption that a character takes 10 on their insight checks on these situations. Beyond that, it assists the DM because calling for insight checks is a sure way to make the PCs realize something is up.
It's the same deal with passive perception, except that's used to stealth rather than deception.
You've missed a key element in a "passive insight" check — they aren't rolled. You compare the character's Passive Insight score to the difficulty of the check, and determine whether they succeed or not based on that. Applying a penalty wouldn't have any effect at all; future "checks" would still fail, only by a greater amount.
Now, you could apply a small bonus if you chose, reflecting greater exposure to the stimulus. But, at that point, you'd be better off rolling.
Generally I would have the players passive insights in front of me and if one of them would automatically be able to tell if somebody was acting oddly or lying, then I would give those players a little extra information.
For example, the basic description: "As you approach, a figure comes up to the bars of the cage and pleads with you to let him free. The figure is a female elf, dressed in tattered rags, and seems to have been beaten."
If a player had a very high passive insight, I might tell that player the additional: "The way in which the figure moves appears unnatural, as if the elf is not comfortable with its body"
I wouldn't have somebody "use" their passive insight. It's more a tool for me to determine how much information to give out without the character making an active check.
From a GMs perspective:
Passive insight is the DC for any subterfuge (bluff & intimidate) checks made against the PC.
It is the PC's defense against social attacks. Much the way AC is the defense against weapon attacks.
If your NPC is trying to lie to a PC, you roll a Bluff of your NPC against the PC's passive insight. If your bluff does not exceed the PC's insight, you should hint to the PC that something the NPC just said does not sound right.
One of the game design goal of things like passive insight (and passive perception) is to allow the DM to adjudicate scenarios where knowledge of the insight (or perception) test would grant the players knowledge that their characters should not.
Imagine a scenario where the characters are approached by an NPC with a story of woe, that in actual fact is a fabrication to try and lure the characters into a trap.
The NPC needs to make a bluff check, but if, as the DM you call for an insight check to oppose the bluff role, you tip your hand to the players that there is something for their characters to discern.
Enter the passive insight score.
As other answers have noted, in order to make this flow most naturally, as the DM you'll want to know the passive insight of your players in advance, and thus avoid having to break out of role play to ask for their characters' stats.
Passive Insight and Passive Perception are based on your Insight and Perception skills. D&D thought it would be helpful to calculate Insight and Perception as passive skills because when NPCs are lying or when enemies are trying to sneak up on you, your DM would benefit from knowing the Difficulty Class of sneaking up or of lying to the whole party. Also perception is used to look for loot, traps or other details when roaming around dungeons or... any place really.
These passive skills are about how PC perceive their surroundings.
In these cases the 'Passive Skills' have a +10 bonus: the +10 'bonus' is actually an average of the roll the game assumes you make. The game assumes then player rolled a d20 and scored a 10. That is the reason of why these 'skill checks' aren't rolled by your players... and from a perspective of D&D being story, this allows more continuity for when a DM is telling you where are you, what do you see, how are the people around you, what are the npcs saying...
It's all about your DM not being to obvious when asking for perception checks or insight checks, you'll know something is going up. These passive skills help with the suspense and pace! Hope this helps!