Can someone explain Lethal Tempo

Hitting a champ will initially give you a small attack speed bonus, but if you continue attacking the enemy champion, the duration is extended.

Damaging a champion gives you the extra attack speed for 3s, and auto attacking them after getting the buff will extend that AS buff duration to 6s.

To add on to what others said, it can literally be any form of damage, even something like a tick of twitch's passive from his thrown cask or a accidental stray Xayah feather poking an enemy laner will set off lethal tempo. Physically right clicking a champion and hitting them with specifically an autoattack will extend the duration of lethal tempo up to 6 seconds. I hope this helps clarify what it does for you a little better