Why are Detectives + Inspectors Colloquially Referred to as Gumshoes?

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It turns out that the original "gumshoes" of the late 1800's were shoes or boots made of gum rubber, the soft-soled precursors of our modern sneakers... At the turn of the century "to gumshoe" meant to sneak around quietly as if wearing gumshoes, either in order to rob or, conversely, to catch thieves. "Gumshoe man" was originally slang for a thief, but by about 1908 "gumshoe" usually meant a police detective, as it has ever since.

The primary modern use of gumshoe is as a noun meaning 'a private detective'. This is a narrow remnant of a group of senses having to do with stealth.

Originally, gumshoe referred to a shoe with a rubber sole, specifically, galoshes or (more relevant to our purposes) sneakers. The senses of gumshoe leading to the 'private detective' sense all have to do with the idea that rubber-soled shoes give the wearer the ability to walk stealthily.

The earliest examples show gumshoe used as an adjective meaning 'being a stealthy or surreptitious thief or tracker', and thus 'being a plainclothes police officer or a private detective'. This adjective is first recorded in 1900 and occurs a number of times before 1910. A secondary sense of the adjective, found at about the same time, is '(of actions or activities) carried out stealthily or surreptitiously' (e.g., "No gumshoe democratic campaign in Nebraska," from a 1904 newspaper).

The noun in the sense 'a plainclothes police officer; (usually) a private detective' is first recorded in 1906, as your source notes. Less frequent noun senses are 'a police officer' and the military use 'an intelligence officer or a spy'.

Some additional uses: the verb gumshoe 'to come or go stealthily; sneak', from 1902; 'to work as a private detective', from 1908; gum, gumfoot, gumboot, and gumheel, all meaning 'a private detective; and gumshoe artist and gumshoer, both also meaning 'a private detective'.