Sometimes when Jesus makes straightforward statements that appear to mean one thing on the surface, they require that we go a bit beneath the surface to resolve the apparent difficulty. In this case, that kind of extra labor is not required. Jesus meant exactly what he said: "The Father is greater than I." That's somewhat distressing for Christians because we have this sacred doctrine of the Trinity that describes the unity of the three persons of the Trinity—Father, Son, and Holy Spirit. Here the Son of God is saying that the Father is greater than he is. This is one of the reasons the church has always confessed a doctrine called the subordination of Christ. Notice that it's not called the inferiority of Christ. I stress that because in our culture some people conclude that subordination necessarily implies inferiority.
The reason Christian theology contains a doctrine about the subordination of Christ is that even though the second person of the Trinity is coessential with the Father (he's of the same essence, "very God of very God," eternal in his being) there is a distinction among the persons of the Godhead. In the economy of redemption and even of creation, we see certain works attributed to the Father, others to the Son, and others to the Holy Spirit.
The traditional view is that the Son is begotten of the Father—not created, but eternally begotten. The Father is not begotten of the Son. The Son is sent into the world by the Father; the Son does not send the Father. Jesus said, "I do nothing on My own authority, only that which the Father tells me to do." His meat and his drink were to do the will of the Father. He was commissioned by the Father to come into the world for the work of redemption. In that plan of redemption in the Godhead itself, one sends the other, and the one who sends is said to be greater than the one who is sent in terms of the economic distinctions and the structure by which the Godhead works.
By the same token, the church historically, except for the filioque dissenters, has stated that, as the Father sends the Son, so the Holy Spirit is sent by both the Father and the Son. As the Son is subordinate to the Father in the work of redemption, so the Spirit is subordinate to both the Father and the Son. But again, that does not mean an inequality of being or dignity or divine attributes. The second person of the Trinity is fully God; the third person of the Trinity is fully God. In that work of redemption we see the expression of superordination and subordination.