The famous Maharshi Atri was the mind born son of Brahma. The rishi was acknowledged as someone who is on par and even greater than his own father Brahma. He was an exalted sage who went beyond the three modes of existence (trigunas - Sattva, Rajas, Tamas) and the three states of experience (jagruth, sushupti and swapna). He was called Atri because he was a trigunaatheetha (beyond the three states). 

He once had a queer desire. He knew that Brahma, Vishnu and Rudra are responsible for creation, sustenance and dissolution of the rahmandas. But there must be someone or something that puts these three (i.e Vishnu, Shiva and Brahma) into action in order to carry on their duties. So, the maharshi went into penance, with the desire to witness the consciousness (paramjyoti) that is the source of the Trimurtis and the entire creation of universe. Anasuya devi, his wife, also performed penance along with him, so that God may shower bountiful grace on mankind. The penance performed by Athri maharshi and his wife went on for thousands of years. 

As the fruit of his penance, Athri maharshi witnessed a bright light (paramjyoti) encompassing the entire universe and that light take the form of a 16 year old child. That 16 year child spoke so, "I am the cause of all causes, the source, and the embodiment of trinity (Brahma, Vishnu and Mahehswara). I am pleased with your penance. I shall take birth as your son." This was the first appearance of Dattatreya that occured within the mental vision of the famous Athri maharshi as result of his tiring penance.  In reality, the child had no name. He was an embodiment of parabrahma in the saguna form. The indescribable and unqualified Brahman without name and form had given itself to the great sage as a child. And hence was called Datta. Datta literally means "that which is given". Since the brahman has gifted itself as a child to the sage Athri, he was called Dattatreya (Datta + athreya = given to Athri). 

Since the trinity - Brahma, Vishnu and Shiva are solely responsible for all activities of existence, anything that comes into being must come via these three only. In reality, the brahman is beyond existence at all levels but even for such all pervading brahman to condense and come into existence within the creation, would imply that he (it) will have to come via Brahma, Vishnu or Shiva only. Hence, the actual birth of Dattatreya took place via the mode of Maha Vishnu and hence the Bhagavatha describes him as the avatar of Maha Vishnu. 

Even before Dattatreya was born, the trinity profusely blessed the couple that the three of them would be born to them as children. The incident behind this boon is another well known story from the puranas. The avatar of Dattatreya is very ancient and even predates the Kurma avatar. According to the puranas, it took place hundreds of crores of years back in a different aeon and different age of time. Later on, Lord Dattatreya assumed various forms and avatars such as the Avadhoota and Anagha. This is a brief description of details that lead to the mysterious birth of Dattatreya.