The mantras are from the Taittiriya Upanishad, Shikshavalli I.20 that says: matrudevo bhava, pitrudevo bhava, acharyadevo bhava, atithidevo bhava. It literally means "be one for whom the Mother is God, be one for whom the Father is God, be one for whom the Teacher is God, be one for whom the guest is God." matrudevah, pitrudevah, acharyadevah, atithidevah are one word each, and each one is a Bahuvrihi samasta-pada.
Tithi in Sanskrit denotes a (calendrical) date. In ancient times, when means of communication were limited and it was not possible for guests to anticipate their date of arrival, atithi (which literally means "without a fixed calendrical time") was coined to depict a visiting person who had no fixed date of arrival or departure. Devah (which, through sandhi or euphonic combination, becomes written/pronounced as devo when followed by certain kinds of consonants) means God and bhava means Be or Is - "be the one for whom the Guest is God".