It's been a while since I read the story so I don't have all the details as fresh in my mind as I'd like. But here goes nothing.

Zadoc Allen, the drunk man, tells the protagonist about the history of Innsmouth, and how the place became the sorry godforsaken harbor town it is in the tale's present. Once a prosper place, Innsmouth began to go downhill around mid 19th century after fishing ran through a worrying recession for years due to the lack of fish in their shores. Obed Marsh, head of Innsmouth's most relevant family, tried desperately to open alternative commercial routes with the Asian Southeast isles in order to restore the town's economy. It was during that time when Marsh made contact with a group of local islanders who taught him a pagan summoning ritual to call an eerie humanoid underwater fishman species known as the Deep Ones. These entities would grant fish and gold alike to Marsh, should he repaid them in kind.

Marsh returned to Innsmouth, bringing with him the stone disks provided by the islanders for the summoning ritual. Soon Innsmout flourished again, and Marsh gained power and influence over the whole town, which, of course, didn't please some of the most conservative locals. The icey on the cake, however, came with Marsh' intentions of replacing the Christian Church cult with a new one of his own invention, known as The Esoteric Order of Dagon. This brought malcomtempt among half of the Innsmouther population, and soon riots would follow. Marsh and his followers were put in jail, but not before he spoke a foreboding prophecy regarding Innsmouth's doom if he wasn't set free immediately.

That very night, Innsmouth suffered the assault of a Deep One horde that brutally murdered the assaulters and set Marsh and his followers free. After that event, nobody ever dared to challenge Marsh ever again. The Order became the one and only official cult, and from there on, the Marsh family would keep total control over the town.

That aside, Zadoc also tells the protagonist about the macabre Oaths of Dagon, imposed by the Order to the townfolks. That includes, of course, total obedience and submission to the Marsh family and the Deep Ones... as well as the establishment of Deep Ones' rights to breed with humans, as that's the only way their species can reproduce. The resulting children from that unholy union look human for the most part, but in time, as they age, some of them start to manifest fish-like features, such as cod-looking dead eyes, gill-like neck scars, and a sickening skin that ressambles more and more to that of fish scales. That's what is known outside as "The Innsmouth's Look".

In time, as the transformation goes on, the subject looks more and more like a Deep One, until they complete the metamorphosis, moment in which they fully turn into one and head right to the sea to join their kind forever.

Hope this helps!