What is the significance of the anointing oil?

The significance of the anointing oil lies in its pointing us to Jesus' finished work at the cross. In the same way olive fruits have to be crushed in order for the oil to flow out, Christ was crushed at different points in His passion so that we may enjoy wholeness in every area of our lives, and that God’s blessings may freely flow toward us.

Interestingly, when olives are pressed for oil in Israel, they go through three rounds of pressing. The first press gives you extra virgin olive oil, which is used in the Jewish temple for lighting. The oil from the second press is used for making medicine (or for healing), while the oil from the third press is used for making soap (cleansing).
In the same way, our Lord Jesus was also pressed three times to provide for us a perfect salvation that includes deliverance from all powers of darkness, divine healing for our bodies and complete forgiveness of all our sins.
His "pressings" began at Gethsemane (meaning "olive press") in a time of darkness, both literally and spiritually. He said to those who had come to arrest Him that night, "…this is your hour, and the power of darkness" (Luke 22:53). The Bible says He was "in agony" (Luke 22:44) as He contemplated what awaited Him from then on to the cross. It was a very dark time for our Lord. He was pressed so that we can be called "out of darkness into His marvelous light" (1 Peter 2:9), and to walk in this light.
Jesus' crushing continued at the scourging post where He was scourged violently by the Roman soldiers. He took lash after brutal lash, bearing every stripe until His skin was torn and His bones exposed (Psalm 22:17, 129:3), all to purchase our healing and health. The prophet Isaiah tells us that "by His stripes we are healed" (Isaiah 53:5).
As the olive is crushed or pressed a third time to yield its cleansing virtue, Jesus' crushing climaxed with His being nailed at the cross. He was crushed under the judgment of a righteous and holy God because He was carrying our sins and being punished for them. He shed His blood at Calvary to wash away all our sins, making us whiter than snow.
So when you use the anointing oil, reflect on how Christ was crushed to bring you out of darkness into His glorious light, heal your body of all diseases and give you total forgiveness of sins. Expect then, to see the benefits of His sacrifice exploding in your life as you use the oil!