Derived meaning: passionately.
It is the word you know, “뜨겁다 (hot).” You've just stumbled upon another sense of the word that is slightly rhetoric: passionately enthusiastic, eager, or excited. It's been derived from the basic sense of the word: having a high degree of heat or a high temperature.
It's more of a word that is used in catchphrases/quotes rather than usual conversations, as is a slightly rhetoric, literary way of expression. Often used in imperative sentences.
And an extra note: considering the usage described above, “뜨겁게 청소하라” sounds a bit weird; why would someone say
“Sweep it all up, with passion.” - Somebody in the world
Or, might be a mistranslation.
It might be a casual, quick, and nonstandard way of saying “뜨거운 물로 청소하라. (Clean it up with hot water.).”
EDIT: turned out this is not the case! See comments below.