Learn Russian: У нас нет остальных чистых тарелок. - We do not have the rest of the clean plates.

Because they are plural adjectives describing тарелок (plates), which is in the genitive case after the negative word нет. In the plural genitive case, regular adjectives take the ending -ых (or -их after certain spelling-rule letters). Since the stems of остальной (remaining) and чистый (clean) end in regular hard consonants, they both take -ых.

The dictionary form for plate is тарелка. Why does the plural genitive become тарелок here?

For feminine nouns ending in -а, the genitive plural usually just drops that final -а. But dropping the -а from тарелка would leave тарелк, which is very hard to pronounce. To fix this, Russian inserts a "fleeting vowel" (о) between the last two consonants, resulting in тарелок.

Why does нет force the words after it into the genitive case?

The word нет literally means "there is not" or "there is an absence of." In Russian, you always use the genitive case to show the absence or non-existence of something. Because of this rule, the entire phrase for "the rest of the clean plates" must take the genitive plural form.

Could I use других instead of остальных in this sentence?