Frodo's parents drowned in a mysterious moonlight boating accident, possibly involving a bottle of wine or ale.
After all his father was a Baggins. A decent respectable hobbit was Mr. Drogo Baggins; there was never much to tell of him, till he was drownded.'
'Drownded?' said several voices. They had heard this and other darker rumours before, of course; but hobbits have a passion for family history, and they were ready to hear it again. 'Well, so they say,' said the Gaffer. 'You see: Mr. Drogo, he married poor Miss Primula Brandybuck. She was our Mr. Bilbo's first cousin on the mother's side (her mother being the youngest of the Old Took's daughters); and Mr. Drogo was his second cousin. So Mr. Frodo is his first and second cousin, once removed either way, as the saying is, if you follow me. And Mr. Drogo was staying at Brandy Hall with his father-in-law, old Master Gorbadoc, as he often did after his marriage (him being partial to his vittles, and old Gorbadoc keeping a mighty generous table); and he went out boating on the Brandywine River; and he and his wife were drownded, and poor Mr. Frodo only a child and all. '
The Fellowship of the Ring: Chapter 1 - A Long-expected Party
Frodo was initially fostered in Brandy Hall (a large multi-roomed residence with enough room for an estimated hundred hobbits) where he lived for nine years until Bilbo offered for him to live in Bag End, ultimately making Frodo his heir, largely because he thought very little of his other relatives.
Mr. Bilbo never did a kinder deed than when he brought the lad back to live among decent folk.
'But I reckon it was a nasty shock for those Sackville-Bagginses. They thought they were going to get Bag End, that time when he went off and was thought to be dead. And then he comes back and orders them off; and he goes on living and living, and never looking a day older, bless him! And suddenly he produces an heir, and has all the papers made out proper. The Sackville-Bagginses won't never see the inside of Bag End now, or it is to be hoped not.'
It might interest you that the part of Drogo was originally cast in The Hobbit: Desolation of Smaug (along with some scenes that explain his absence in the Lord of the Rings trilogy) but the scene was ultimately abandoned before filming took place with the actor being recast as Alfrid.
One of those questions might have been ‘Who is Drogo Baggins?’, for that was the part Ryan [Gage] was offered. To be fair, the answer would not have been found in the pages of The Hobbit, but in the opening chapter of The Fellowship of the Ring where the patrons of The Ivy Bush are discussing Frodo’s ancestry. ‘His father was a Baggins,’ Gaffer Gamgee tells his cronies. ‘A decent respectable hobbit was Mr Drogo Baggins; there was never much to tell of him, till he was drownded.’
The tragic death of Drogo Baggins (resulting in Frodo being orphaned and adopted by Bilbo) was, for a time, featured as part of the prologue in the script. But Drogo, it seems, had been written into the story in order to find a role for the young actor who had made such an impression on the producers; when the scenes featuring Drogo were later abandoned, Ryan found himself being reconsidered for Alfrid, the role for which he had originally auditioned.
The Hobbit Desolation of Smaug - Official Movie Guide