In the Ancient Greek language (not sure about Modern Greek) vowels were either aspirated or not. Essentially that means that whether or not there was a "rough breathing sound" or an H sound before a word. Modern typography uses marks that look similar to commas. For instance the word hoplites should be written in Greek as ὁπὶιτες with the mark over the ο as a marker of aspiration. The word for horse, ἱππος, also uses that marker whereas the name for Italy would be Ἰταλια with the mark reversed.

So to some it all up, there is technically an h sound in those words it's just that Greek didn't have a letter for it, whereas English does so we add the H when we transliterate Greek words with an aspirated vowel.