This idiom is used to make polite requests:
कुछ खाना खाया जाए? – “Should we eat some food?”
This sentence is literally “Shall some food be eaten?”. The implied agency is “by us”. Thus, this implies a first person plural, inclusive question.
थोड़ी चाय पी जाए? “Should we drink some tea?”
कुछ केले खाए जाएं? – “Shall we eat some bananas?”
This idiom is also used to make exhortations. Usually the intonation in the speaker’s voice differentiates an exhortation from a question. A question has a rising intonation, whereas an exhortation has a normal intonation.
कुछ खाना खाया जाए – “Let’s eat some food” (Exhortation)
कुछ खाना खाया जाए? – “Shall we eat some food?” (Question)
Note that the verb agrees with the subject, so जाएं was plural. खाए was not plural because perfect participles are not pluralized unless they are the final component of a verb form.
This idiom is very common in Hindi.