It is "one and a half hours" because you have more than one hour ( one hour plus half an hour ) hence implying that it is plural and not just a single hour. both "one" and "a half" are describing the noun "hour", so two. the other way that it could be stated is "one hour and a half".
Just so, why is it half an hour instead of half a hour?
The initial sound of hour is /a?/, a vowel (well, actually, a diphthong), and not the consonant /h/. Hour is pronounced the same way as our - we say the h is unaspirated. The 'rule' is to use an rather than a before a word beginning with a vowel sound.
What time is half an hour?
a period of 30 minutes. the midpoint between the hours: The clock struck on the half-hour.
Is it in an hour or a hour?
Yes. If a word begins with a vowel sound, then the correct article is an; otherwise, if it begins with a consonantal sound, the correct article is a. Because hour is typically pronounced with a silent h, an hour is correct.