食べさせる = causative form of the verb. As nobu said, the causative form means "make/let (someone) do"
To form a causative verb, for る-verbs you take the stem and add させる (or more casually, さす). For う-verbs, you change the last vowel to あ (or わ, as with the negative form) and add せる (or more casually, す). So 飲む -> 飲ませる, 買う -> 買わせる.
For the irregulars, する -> させる and 来る -> こさせる.
In this case, 食べさせてください would mean "please let me eat it" (though the specific translation could depend on the context).
