Who is it that cries 'Abba, Father' — us or the Spirit in our hearts?

Believers cry 'Abba, Father,' not the Spirit as a separate speaker. The Spirit of God's Son is sent into the hearts of every believer, and through His unction—His enabling work—the believer cries 'Abba, Father.' So the cry is ours, but it is prompted and empowered by the indwelling Holy Spirit.

Les teaches that because of the believer's position after faith, God is the one who precipitates the action. He has sent forth the Spirit of His Son into the hearts of every believer—not into a denomination or a church building. You do not pick up the Spirit by walking through a door on Sunday morning. The Spirit is in the heart of every believer. So when the text says the Spirit cries 'Abba, Father,' it clarifies, 'that is, we cry, through the unction of the Holy Spirit.' The cry is ours, but it is prompted and empowered by the indwelling Spirit.

Galatians 4:6

And because ye are sons, God hath sent forth the Spirit of his Son into your hearts, crying, Abba, Father.

How are believers able to call God Abba, Father?

Believers can call God Abba, Father because at salvation God sends the Spirit of His Son into their hearts. This privilege rests not on human merit but on the finished work of the cross. Through the Holy Spirit’s leading and guiding, born-again believers have every right to approach God as Father.