"When the Counselor comes, whom I will send to you from the Father, the Spirit of truth who goes out from the Father, he will testify about me."
One of the Holy Spirit's main functions is to testify about the identity of Jesus Christ. However, he does not testify with persuasive words or clever arguments, but with a demonstration of power so that faith might not rest on men's wisdom, but on God's power. (1Cor 2:4) It is the Holy Spirit which witnesses the world that Jesus Christ is the Son of God and without him we cannot testify.
In the book of Acts, we see the Holy Spirit testify through men such as Peter, John, Stephen, and Paul. He fills us and enables us to speak the word of God boldly (Acts 4:31) so that we can be, as Christ commanded, "his witnesses in Jerusalem and Samaria, and to the ends of the Earth." (Acts 1:8)
The power of the Holy Spirit is the power to testify through the Fruit of the Spirit: love, joy, peace, patience, kindness, goodness, gentleness, and self-control. These things are not of this world are are signs that there exists another world which they are from. This world is filled with rage, pain, selfishness, lewdness, drunkenness, death and immorality but the Holy Spirit testifies there is life to be had outside of this world.