What is the meaning of this sentence from "A Study in Scarlet" by Sir Arthur Conan Doyle?

I would submit that the first part of the statement is a self-justification for indulging in the life-style of the idle. He stipulates that he has no purpose in life. Why not join the flow?

The author is saying that as a free person with minimal responsibility, he was attracted to London with all the good and bad adventures London had to offer, but hinting at the darker side of life.

gravitated to London

means attracted to, drawn to, the author is saying these types of people naturally and move towards this destination (London) just as gravity is a natural (an irresistible) force

that great cesspool

A cesspool is a pool (or pond) of human waste (both solid and liquid), referring to something as a cesspool means it has undesirable qualities, in this case there is a moral context

loungers and idlers

refers to types of people that are socially undesirable, those that do nothing and do not contribute to society

of the Empire

refers to the once mighty English Empire(commonly referred to post the Acts of Union as the British Empire) of which London was (and still is) its capital and largest city

irresitibly drained

Draining is a process driven by gravity, "draining water from a bathtub". It means the undesirable people are removed from other parts of the geographically spread Empire and drained into (or concentrated into) London. In this usage irresitible means can not be stopped.

The author is saying that London, being the major city of the Empire, attracted many different types of people especially undesirable ones, as any major city in the world still does. Please keep in mind this is a picture of a Dickensian London. The author's attraction for adventure is as a moth to a flame.

There are two metaphors here.

Metaphor One

The narrator had no reason to settle down anywhere: he had no family and no friends. He had a small income that allowed him to spend as much time as he liked without looking for a job. He had nothing to do. The implication here is that he was, at that point of his life and career, a drifter: moving around without purpose, like a rogue asteroid in interstellar space.

A large space object - like a planet or a star - can attract such an asteroid, which will thereupon gravitate towards it. In the British Empire, London is that planet (or star) for all drifters.

Metaphor Two

The second metaphor compares London to a cesspool: an underground storage space into which all kinds of waste is drained. Loungers and idlers are being compared to waste drained into a cesspool, and London to a cesspool. This view is complimentary neither to idlers nor London, but it was very popular among England's upper middle class towards the end of the 19th Century: the very group that was to constitute the vast majority of Sir Arthur's readers.