People who manage parse the sentence also aren't trying understand it, except as far as "Buffalo" is a proper noun denoting a city, which can be used to form then phrases "Buffalo buffalo" == buffalo of/from/belonging to/related to Buffalo, and trying various combinations of interpreting "buffalo" as a noun (in various roles as subject, direct object and so on) or verb, and determining elided words such as "which" or "that" complementizers heading off phrases and embedded clauses.

It's almost purely syntactic reasoning. Searching these spaces of possibilities is something which, you would think, a "natural language parser" ought to be doing to earn its name.

Nobody actually knows what it means "to buffalo" something; it is not necessary to know. People solve the parse in spite of knowing that there is nothing to understand in the sentence.

"buffalo" can mean something like "bother" as an English verb (at least in American informal use), so the whole sentence as parsed in English does have a concrete mental image associated with it, in case that makes any difference.