You left out the part of the passage which speaks of Jesus as having been "in the form of God" before the Incarnation. This makes all the difference in the world in understanding what Paul is writing about.
Apparently the apostle is quoting a primitive Christian hymn contrasting Adam and Christ. Adam, who was the "image of God" but not equal to God, tried to be equal to him (Gn 3:5). Christ, who was in the "form of God" and thus equal to God, didn't think the prerogatives accompanying this equality were something to be "tenaciously held on to" (the Greek word used by Paul is harpagmon), but surrendered them to take on human nature and die on the cross (Phil 2:7-8).
Paul goes on to note how God has bestowed on Jesus "the name that is above every name" and how "at the name of Jesus every knee should bend, of those in heaven and on Earth and under the Earth, and every tongue confess that Jesus Christ is Lord, to the glory of God the Father" (Phil 2:9-11).
This is a paraphrase of Isaiah 45:23, where God says, "To me every knee shall bend; by me every tongue shall swear." By applying this to Jesus, Paul isn't undermining the divinity of Christ, but acknowledging it.