DIAGNOSIS This is almost always a gasket or o-ring issue, not a tightening problem. The included rubber gasket is either missing, installed incorrectly, or got pinched during setup.
FIX First, turn off the water and unscrew the filter unit from the faucet adapter. Check the black rubber gasket inside the adapter's threaded collar. Make sure it's seated flat in its groove, not twisted or folded. If it looks good, reinstall it and hand-tighten the unit—don't use tools or extreme force. Turn the water back on slowly.
IF THAT DIDN'T WORK The gasket might be defective or the wrong size. PUR includes a spare gasket in the box. Swap it in. If you lost it, you can order a replacement gasket kit (often part #RFGPK) directly from PUR.
HEADS UP Overtightening can actually warp the plastic adapter or damage the gasket, causing the leak you're trying to stop. It only needs to be snug enough to stop drips when the water is off.