The German word for “to exaggerate” is “übertreiben”. Literally, this word translates into English as “to over-push” or “to over-drive” (as in a herd of cattle that is driven too far):

“Er übertreibt wenn we sagt, daß er der beste Lehrer in Deutschland ist.” – “He’s exaggerating when he says he’s the best teacher in Germany.”
“Das ist total übertrieben.” – “That’s totally exaggerated.”
“Das ist eine Übertreibung.” – “That’s an exaggeration/overstatement.”

Interestingly, in order to express the opposite of hyperbole, Germans can simply say “untertreiben” (“to understate”) or “Untertreibung” (“understatement”).

PS: In case you were curious, the English “exaggerate” comes from Latin and means “to pile up”.