Why does the devil, the enemy of our souls, want to keep us from obeying the gospel?

Because when we obey the gospel we have forgiveness of sins. The devil no longer has the power of death over us.

"Forasmuch then as the children are partakers of flesh and blood, he also himself likewise took part of the same; that through death he might destroy him that had the power of death, that is, the devil; And deliver them who through fear of death were all their lifetime subject to bondage." - Hebrews 2:14-15 KJV

By Jesus’ death, burial, and resurrection, the gospel, Jesus destroyed the devil and his power of death. (Heb.2:14-15, 2Tim.1:10, Eph.1:13-14, Rom.1:16)

How do we obey the gospel? By faith in Jesus we repent, becoming dead to sin. We are buried in water baptism in Jesus name for the remission of sins. We receive the gift of God’s Spirit, the Holy Ghost, to live a new, holy life. (1Cor.15:1-4, Rom.6:1-4, Heb.4:2, 1Pet.4:17, 2Th.1:3-12)

"Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost." - Acts 2:38 KJV

We are justified [made righteous in God’s eyes] by the name and blood of Jesus. We are sanctified [made holy, separated unto God] by His Spirit, the Holy Ghost. We are redeemed [set free from wrath by Jesus paying the price] by His blood. Our sins are forgiven. (1Cor.6:11, Eph.1:7, Col.1:14)

"But if our gospel be hid, it is hid to them that are lost: In whom the god of this world hath blinded the minds of them which believe not, lest the light [God’s mercy, forgiveness] of the glorious gospel of Christ, who is the image of God, should shine unto them." - 2 Corinthians 4:3-4 KJV

"To open their eyes, and to turn them from darkness [the wrath of God] to light [the mercy of God], and from the power of Satan unto God, that they may receive forgiveness of sins, and inheritance among them which are sanctified [made holy, separated unto God] by faith that is in me." - Acts 26:18 KJV

After we have obeyed the gospel, the devil can no longer accuse us of the sins we have repented of - they have all been washed away by the blood of Jesus. (Rom.1:16, Rev.12:10, Rev.1:5)

"And the God of peace shall bruise [crush] Satan under your feet shortly. The grace of our Lord Jesus Christ be with you. Amen." - Romans 16:20 KJV