The plural form of the word "moose" is also "moose." The word "moose" is an invariant, meaning the word does not change from its singular to plural form. Other examples of invariants include sheep and deer.
Much of the confusion over the plural of "moose" relates to the plural of "goose." Because the two words rhyme and have similar spellings, it makes sense to assume that the shift from singular to plural is parallel. Using that logic, a person may assume the plural of "moose" is "meese," just as the plural of "goose" is "geese."
"Moose" and "goose" have distinctly different origins, however. "Moose" is a loan word, which came to English from the Native American Algonquian language. The plural form comes from the donor language as well. The word "goose" is an Old English word, and the plural form "geese" reflects a convention of Old English language in practice at the time the word was formed.