"As the feast of Easter developed in Christian tradition, so did the festival's preparatory period, known as Lent. [Lent] involved fasting and later abstinence from certain foods, including eggs." We have a record as early as A.D. 330 of Lenten fasts in the church which included eggs.
"Presenting gifts of eggs at Easter has a long and culturally diverse lineage[, especially in areas dominated by the Orthodox tradition]. Practicality was one factor: Given that hens would be laying eggs throughout Lent, a surplus would exist by Easter, probably at lower prices."
The coloring of eggs is linked to various legends, and the color most often used early on was red for the blood of Christ.
The full article from christianhistory.net is found here.