It is difficult to determine the source of the leak from your pictures, water leaking down the chrome faucet body is almost impossible to see without careful scrutiny; it could be coming from one or more of several components.
I will address a likely cause and help you on your way to disassembly and troubleshooting.
When the spout swings around it does so while riding on two rubber o-rings. Water enters the spout from the main mixing valve and these o-rings prevent that water from leaving the spout except for at the sprayer head. The o-rings get worn flat over time, and the seal is compromised, they will be located near the top and bottom of the spout body:
To remove the spout and change the o-rings, you must remove the handle and the retaining cap. Start by using a small flat screwdriver to remove the trim plug that hides the handle retention screw:
There will likely be a set screw in there, possibly requiring a hex wrench. Remove the handle, the retaining cap, then pull up on the spout to remove it. I would suggest replacing all seals, o-rings, valve cartridge at this time. And get some waterproof plumber's grease, it will improve longevity of the spout o-rings and make your spout swivel nicely.