This is simply a tradition, so you will be unlikely to find an explicit reference for it. As mentioned in the other answers, ensuring the player and all other participants know who the caution or send-off signal is intended for is important, as is ensuring any player on the ground is not injured.
What has not been mentioned is that for cautions, it is desirable to elicit a change in player behaviour, since they will usually be remaining on the field of play after the caution. When a player is cautioned, the referee can save themselves a potential sending-off for a second caution by trying to calm a player down and outlining to a player that they are on their final chance.
Being cautioned with a (usually menacingly tall - most elite-level football referees are well over six feet tall) referee standing over you can be an intimidating and humiliating experience, and is unlikely to elicit a positive response in which the player listens to the referee. Therefore, the player is asked to stand - or the referee may offer them a hand up, so they are on equal footing with the referee and so that the interaction is respectful.
The referee then explains what the caution is for, and if they feel that the player is potentially in an agitated or irrational state and likely to reoffend, reminds them that if another offence of a similar nature occurs, they will have no choice but to send them off and that they would much rather that the player stays on the field. If required by the competition rules, the referee may also ask the player's name during this stage - in most competitions, the number suffices and the name is taken from the match sheet.
Such an interaction is not possible if the referee rushes in, stands over a downed player and flashes the card. Even in heated situations, a referee can at least sprint to the location of the downed player, with the card in hand without showing it. This way, other players know that punishment is coming and they should not retaliate, yet the referee still has the opportunity to attempt to improve the player's future conduct.