The meaning of "altura" that you learn in school is "height." But this word has an unexpected meaning: it also translates as "dignity."
Se lo dijo con mucha altura means "She told him in a very dignified manner."
Reaccionó con mucha altura means "He reacted with great dignity."
Note that "height" would make no sense as a translation here, and also note that altura does not mean "haughtiness," even though that would be a reasonable guess; the Spanish word for "haughtiness" is altanería.