[spoiler=Try it on your own, first. Then check out wikipedia...']The sentence is unpunctuated and uses three different readings of the word "buffalo". In order of their first use, these are
* c. The city of Buffalo, New York (or any other place named "Buffalo"), which is used as an adjective in the sentence and is followed by the animal;
* a. The animal buffalo, in the plural (equivalent to "buffaloes"), in order to avoid articles and is used as a noun;
* v. The verb "buffalo" meaning to bully, confuse, deceive, or intimidate.
Marking each "buffalo" with its use as shown above gives
Buffalo/c buffalo/a Buffalo/c buffalo/a buffalo/v buffalo/v Buffalo/c buffalo/a.
Thus, the sentence when parsed reads as a description of the pecking order in the social hierarchy of buffaloes living in Buffalo:
[Those] (Buffalo buffalo) [whom] (Buffalo buffalo buffalo) buffalo (Buffalo buffalo).
[Those] buffalo(es) from Buffalo [that are intimidated by] buffalo(es) from Buffalo intimidate buffalo(es) from Buffalo.
Bison from Buffalo, New York who are intimidated by other bison in their community also happen to intimidate other bison in their community.
It may be revealing to read the sentence replacing all instances of the animal buffalo with "people" and the verb buffalo with "intimidate". The sentence then reads
"Buffalo people [whom] Buffalo people intimidate [also happen to] intimidate Buffalo people."
Preserving the meaning more closely, substituting the synonym "bison" for "buffalo" (animal), "bully" for "buffalo" (verb) and leaving "Buffalo" to mean the city, yields
'Buffalo bison Buffalo bison bully bully Buffalo bison', or:
'Buffalo bison whom other Buffalo bison bully themselves bully Buffalo bison'.