Learn Russian: Где ключи от остальных комнат?

It is the genitive plural form of остальной (other or remaining). The preposition от forces the genitive case, and since we are talking about plural rooms, the adjective must be genitive plural too. The standard genitive plural ending for adjectives is -ых.

Why is the preposition от used here when the English translation says keys 'to' the rooms?

In Russian, a key is thought of as belonging 'from' a door or room, rather than going 'to' it. Therefore, you always use от (from) followed by the genitive case when talking about the key to something.

Why is the word for rooms spelled комнат here with no ending?

комнат is the genitive plural of the feminine noun комната. When a regular feminine noun ends in -а, its genitive plural is usually formed by simply dropping that -а, leaving what is called a zero ending.

Why is the plural of ключ spelled ключи and not ключы?

This is a perfect time to remember the 7-letter spelling rule! After the letter ч (and six other specific consonants), you must always write и instead of ы. This rule applies when making this masculine noun plural.