I own one of these Brita 10-cup water filter pitchers, and found it a bit tricky to take it apart to clean the inside. Buried in a review on Amazon were these great instructions:
How to take off the filter housing is quite simple: fill the pitcher with soap and water, flip it upside down and the housing should just slip off.
The important part is to fill up the bottom of the pitcher, leaving the top empty. When you flip the pitcher upside down, the weight of the water will make the white plastic insert pop out.
Update: This eventually became too hard, since the insert can stick, and make it hard to remove. I bought a ZeroWater pitcher and am quite happy with it, it’s easy to disassemble for cleaning.