He is being obedient to God the Father.
A related Scripture here is Romans 5:19, where Christ is placed in contrast to Adam:
For as by one man's disobedience many were made sinners, so by the obedience of one shall many be made righteous.
John Chrysostom's discussed this passage in the 4th century in the context of the Arian controversy (a dispute over the divinity of the Son):
See, says one, He voluntarily became obedient; he was not equal to Him
whom He obeyed. O ye obstinate ones and unwise! This doth not at all
lower Him. For we too become obedient to our friends, yet this has no
effect. He became obedient as a Son to His Father; He fell not thus
into a servile state, but by this very act above all others guarded
his wondrous Sonship, by thus greatly honoring the Father. He honored
the Father, not that thou shouldest dishonor Him, but that thou
shouldest the rather admire Him, and learn from this act, that He is a
true Son, in honoring His Father more than all besides.1
A more recent commentary connects the passage also with the Gospel accounts of the Agony in the Garden:
From the beginning He made it plain that He would obey the Father's
will, which was His own in His divine nature, even to the point of
keeping the whole law "to fulfill all righteousness" (Matthew 3:15);
"I seek not mine own will, but the will of the Father which hath sent
me" (John 5:30). Just before His crucifixion, He prayed, "O my
Father, if it be possible, let this cup pass from me: nevertheless not
as I will, but as thou wilt" (Matthew 26:39); "For I came down from
heaven not to do my own will, but the will of Him that sent me" (John
6:38).
The Apostle stresses our Lord's perfect obedience: "Though He were
Son, yet He learned obedience by the things which He suffered"
(Hebrews 5:8); "and being found in fashion as a man, He humbled
Himself, and became obedient unto death, even the death of the cross"
(Philippians 2:8).2
1 Homily VII on Philippians
2 Dmitry Royster, St. Paul's Epistles to the Romans: A Pastoral Commentary, p.139