The caracal is built around its hind limbs: long tibia and femur, heavy thigh musculature and a light frame. That geometry converts into a very high standing jump, and the cat uses it to intercept birds rising from the ground rather than pursuing them.
Eyewitness and photographic accounts describe caracals batting down doves and guineafowl mid-flight, occasionally striking several in a single leap. Historical sources from India and Persia do record captive caracals being kept for hunting and set against penned birds as a wager. The step from that practice to the English idiom is folk etymology rather than evidence: the phrase is already recorded in English by 1706, explained there by the plain image of a cat loose in a dovecote, and no source connects it to caracals.
The leap heights repeated in popular accounts, commonly given as around three metres, come from observation and estimate rather than measured trials, so they should be read as approximate. The equally common assertion that the caracal is a lynx is outdated: despite the ear tufts, it belongs to its own genus in an African lineage alongside the serval and African golden cat.