DIAGNOSIS That's almost always a seal problem. The silicone gasket that runs around the edge of the filter lid is either misaligned, has some grit on it, or has gotten stiff and isn't sealing properly.
FIX First, take the whole filter lid assembly off the pitcher. Flip it over so you're looking at the underside. You'll see a thin, clear or white silicone gasket seated in a groove. Carefully pop it out with your fingernail. Give it and the groove a good rinse under warm water to wash away any mineral grit or debris. Make sure the gasket isn't twisted or pinched anywhere, then press it firmly back into its groove all the way around. When you reattach the lid to the pitcher, listen for a solid, satisfying click on all sides.
IF THAT DIDN'T WORK Inspect the gasket closely for any tiny nicks, tears, or permanent flat spots. If it's damaged, it needs replacing. You can order a replacement gasket/seal kit directly from Brita's website or through their customer service. It's a cheap part.
HEADS UP Don't overtighten or force the lid on thinking it'll seal better. It's designed to click on with moderate pressure. Cranking on it can actually warp the plastic or further dislodge the gasket, making the leak worse.