From what I understand, "食べましょう" and "食べよう" both imply, more or less, the same meaning, which is 'to invite someone to eat with them', the latter being a verb conjugation of the volitional form, the more casual 'ましょう'.
However, "食べる", the base verb which means 'to eat', is much more commonly used for the invitation message. I would like to know why that is so, and when have people started to use it, so much so, that is has become a colloquial phenomenon.
Thank you.