The short answer
Your hose bib drips after you shut it off because the part inside that's supposed to seal (a rubber washer) has worn out, or because a hose left attached is trapping pressure and forcing water out the vacuum breaker on top.
Summer doesn't create the problem so much as expose it. You're using the faucet constantly, and a washer that limped through winter finally gives up.
Before you spend a dime, do one thing: unscrew the hose, open its nozzle to bleed the pressure, and watch the bib for a minute. If the drip stops, you were never dealing with a broken faucet at all.
What's actually going on inside
A hose bib is a simple valve. You turn the handle, a threaded stem screws down, and a rubber washer at the bottom presses against a metal seat to stop the flow. When it drips, one of those two surfaces isn't sealing.
The washer is the usual suspect. Rubber gets hard and flat with age and heat, and once it can't fully compress into the seat, you get that maddening drip that no amount of handle-cranking fixes.
The seat is the second suspect. On hard water, mineral scale and grit slowly pit and groove the metal. When the seat's rough, even a brand-new washer can't make a clean seal.
If the water shows up around the handle instead of the spout, that's the packing, a separate seal that keeps water from creeping up the stem. It usually just needs the nut snugged.
And if you've got a frost-free sillcock, remember the working parts are at the far inside end of that long tube. The stem you pull out will be 8 to 12 inches long, and the washer lives at the tip.