What does Rarest mean in the Badges section of your Profile tab?
To whom are they indicating to?
Rarest = Not many other users have this badge = Maybe a lot of other people have this badge, but it's still less people than how many people have these other badges
So the reason that they show the rarest badges is because most users would want to know which badge(s) they have that most other people haven't been able to earn. Same goes for viewing other people's profiles.
The more a badge is given out, the less "valuable" it can be seen as in SE's "badge marketplace". So the "rarest" of the rarest badges are seen by most people as the most valuable. Since they are the "toughest" to earn.
It is the least awarded badge (to others) on that specific site that you have.
For example, if you have
Legendary, on almost every site, that badge will not be given very frequently. See the list of badges:
Legendary has been awarded only seven times!
The badges you see there should let you appreciate the particular badge a little more, as there are just a few people who have earned it.