Sly and Bentley gained the ability to crawl in Sly 2. It is used to squeeze through tight spaces, allowing escape or access to otherwise unreachable areas, often both.
Crawling was first used in the mission "Breaking and Entering" in "The Black Chateau" (unless used prior in free-roaming), first by crawling underneath tables to evade flashlight guards, and then through two air systems.
Sly is able to use this ability in every game since Sly 2. Bentley loses the ability to crawl due to being in a wheelchair starting from Sly 3. Murray, the Panda King, and Bob Cooper are incapable of crawling into tight spaces due to their large size. Carmelita Fox also cannot crawl into tight spaces, but the reason for this is unknown. It is unknown if Sir Galleth Cooper is capable of crawling into tight spaces, as tight spaces never appear in any of the areas where he is playable.