Origins
Installing software is a secular echo of a sacred act. The word comes from Medieval Latin installāre — 'to place in a stall' — from in- ('into') + stallum ('a seat, a stall'). The original installation was religious: a bishop or abbot was installed by being formally placed in their cathedral stall, the carved wooden seat that was their symbol of authority.
The stallum itself has Germanic roots. Frankish *stall meant 'a standing place' or 'a position', from Proto-Germanic *stallaz. The same root produced English stall (a market booth, a stable compartment, a church seat), stallion (a horse kept in a stall for breeding), and forestall (to block someone's standing place — to prevent them from acting).
The ceremonial meaning dominated for centuries. Medieval installation ceremonies were elaborate affairs: the new bishop processed through the cathedral, took their oath, and was physically guided to their stall by attending clergy. The act of sitting down in the correct seat was the installation.
Later History
The mechanical sense — fitting equipment into position — appeared in the 19th century as industrialisation created a need for the concept. Installing a steam engine or a telephone switchboard carried the same logic: placing something in its fixed, proper, designated position.
The computing sense, first documented in the 1960s, extended the metaphor naturally. Software is installed by placing it in its designated location on a system, where it can operate from its assigned seat.