They're called (for obvious reasons) boat booms. The gear attached to them varied somewhat, but you can see most of it in that excellent photo. There's a line running up from the tip of the boat boom to the end of the lower yard, a simple, two-part tackle leading to the bowsprit, and another line (barely visible on the starboard side) leading somewhere aft. Those three keep the boom from swinging violently in any direction. There's also a rope ladder hanging down from the end, and a footrope beneath the boom. To board a boat tied up at the boom, a man would make his way out along the boom by way of the footrope and climb down the ladder to the boat.
The inboard end of the boom generally would be secured to the hull permanently by means of a gooseneck-like pivot mechanism. When the ship got under way, the boom would be swung aft, and the outer end of it would be lashed to a simple crutch fitting. Most of the gear would be removed (though I think I've seen some photos from this period showing the ladder rolled up next to the boom).
In ships fitted with studdingsails, the boat boom frequently did double duty as the fore lower studdingsail boom, spreading the foot of the lower studdingsail. It's pretty clear, though, that the Kearsarge didn't have studdingsails.
The gear in the photo is, in fact almost identical to the boat boom gear used in warships as recently as World War II. I'm not sure how many, if any, warships still handle their boats that way.
Hope that helps a little. Good luck.
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