Personal rating was added to the in Arena patch 2.4.2 in order to deter team switching as well as make it harder for people to abuse the system and trivialize the rating requirement on certain pieces of gear.
In order to buy Arena gear that requires a certain rating, you must have both your personal and team rating at the proper level.
When you leave a team and join a new one, your personal rating resets to 1500, regardless of the team's rating.
If any of the team's members currently playing has a personal rating that is 150 points higher or lower then the Team Rating, then the matchmaking system averages the personal rating of all those playing and queues the team based on this new value.
Example: A 3v3 Team has a Team Rating of 2,000. They get a new member in and play some games. The new member's Personal Rating is 1500, way below the 150 range. The Average rating is 1833.33 ((2000+2000+1500)/3) - so the team will be Queued against teams in that average ranking.
The effect of this is that while the lower member's Personal Rating will go up/down at a normal rate, the Team Rating and Personal Rating of the rest of the team will go down/up at a much different rate. A 2000 team losing to a 1800 team loses a lot of points, while winning is worth almost nothing.
In this way, Blizzard tries to discourage constant team switching and people joining a team simply to be "carried" to a certain threshold.