Essentially it makes your avatar appear offline to scripts and allows you to appear truly offline to anyone that is not right beside you in world. This will allow you to avoid malicious sensors, and prevent people from interacting with your avatar when you don't want to be interacted with (for example you're busy).

It will do the following:

• Make you invisible to sensors, unless you are the owner, or on the owner's land (so security scripts can still see you).
• Make you always return an offline status to llGetAgentData() calls, unless you are the owner, or on the owner's land (though why it'd matter is beyond me).
• Make you always return null values to llGetObjectDetails() calls using your avatar key, unless you're on the owner's land.
• Make you always return false, or an error to calls to llGetAgentInfo() unless the script is owned by you, or you are on the owner's land.
• Causes you to ignore attempts to get permissions from you (e.g to animate you), unless you are the object owner, or on the owner's land, or sitting on the object (in which case permissions are automatically given as normal).
• Causes you to ignore dialogues from objects, unless you are the object owner, or on the owner's land, or sitting on the object (in which case permissions are automatically given as normal).

There may be some specific scripting functions I missed, but you get the idea, basically the scripting virtual machine will be aware of your invisible status for ALL functions that can process an avatar's key.

It essentially makes it impossible for follower scripts and similar annoyances to follow or target you in any way.

In addition to this, the option will enable more robust IM/chat ignoring functions in combination with "Busy" mode. i.e. - if you set yourself to privacy mode, then any IMs you receive will be replied to with the "User offline" message, unless the user in question is on your friends list and always allowed to see you. You will receive chat as normal.
If you are in privacy mode AND busy, then your viewer will queue any IMs you receive and reply with the user offline message to people, or in the case of friends who are always able to see you, they will receive your busy message. All messages will be queued somewhere for you to see when you remove busy status.

As for combat/gaming/roleplay sims where this could be harmful to some functionality, there would be an estate-level function to allow land-owners to prevent privacy mode on their land. Easiest way would probably be to block "private" avatars as with non-paying or non-verified avatars, but to prompt the user with a message telling them they must disable privacy mode to enter the parcel. Otherwise the more difficult (but better) way is to add two flags to an avatar, one indicating if they have set themselves to privacy mode, and another to indicate whether they actually ARE in privacy mode. This way when you pass into a parcel that disables privacy mode, then the setting is turned off, but when you leave it is turned back on if you want it on.