The eagle is one of the four living creatures. Ezekiel 1:10 describes four faces on each creature, a man, a lion, an ox and an eagle, and Revelation 4:7 gives four separate creatures with those forms around the throne. The pairing of the four with the four gospels is a Christian development, and it is neither original nor uniform.
Irenaeus, writing around 180, is the earliest to make the analogy in Against Heresies 3.11.8, and his assignment is not the familiar one: he gives the lion to John, the ox to Luke, the man to Matthew and the eagle to Mark. Augustine offered another arrangement again, with the lion to Matthew and the man to Mark. The scheme that won is Victorinus of Pettau's, man to Matthew, lion to Mark, ox to Luke, eagle to John, which Jerome adopted, and which passed through Gregory the Great and works like the Book of Kells into the standard Western iconography.
The reasoning for the eagle and John is that his gospel opens in heaven rather than on earth, with the Word rather than a genealogy or a wilderness. The reasoning is retrospective in every case; the assignments were made to fit and then justified.
That there are at least three competing patristic schemes is worth stating, because the familiar arrangement is usually presented as though it were self-evident or apostolic. It is neither. It is a fourth-century editorial preference that succeeded.
Garuda, the eagle-like mount of Vishnu, is a separate tradition with no relation to any of this. He is the enemy of serpents, the vehicle on which Vishnu flies, and a state emblem of Indonesia and Thailand. The visual resemblance to Western imperial eagles is convergence: a large raptor is what a state reaches for when it wants to look formidable.