Here is an explanation of the feast, from the Catholic Encyclopedia:
"The object of this feast is to commemorate the dignity of the Mary as Mother of God. Mary is truly the Mother of Christ, who in one person unites the human and divine nature. This title was solemnly ratified by the Council of Ephesus, 22 June, 431. The hymns used in the office of the feast also allude to Mary's dignity as the spiritual mother of men. The love of Mary for all mankind was that of a mother, for she shared all the feelings of her son whose love for men led Him to die for our redemption (Hunter, Dogm.Theo. 2, 578). The feast was first granted, on the petition of King Joseph Manuel, to the dioceses of Portugal and to Brasil and Algeria, 22 January, 1751, together with the feast of the Purity of Mary, and was assigned to the first Sunday in May, dupl. maj. In the following year both feasts were extended to the province of Venice, 1778 to the kingdom of Naples, and 1807 to Tuscany."
The feast was made universal in 1931, in honour of the fifteenth centenary of the Council of Ephesus. It was shifted to January 1 in the new calendar, allegedly for ecumenical reasons (though it is not particularly obvious what these are).
In any case, for a nice meditation on what it would have been like for Our Lady to be pregnant with Our Lord, have a read of this post over at New Theological Movement (from whence cometh the icon).