“This Jesus did God raise up, of which we all are witnesses. Being therefore by the right hand of God exalted, and having received of the Father the promise of the Holy Spirit, he has poured forth this, which you see and hear.”
Having given his Scriptural proof Peter now applies it powerfully. This Coming King was Jesus, and Him therefore God has raised up, as all His disciples present had witnessed. And having been raised up He has ‘poured forth' His Holy Spirit. (The Spirit was thought of as ‘poured forth' because He was thought of by the prophets as like pouring rain). And this present pouring out of the Spirit is proof positive that He has been exalted by God's powerful right hand (compare Acts 2:25). For the coming of the Holy Spirit, with the manifestations that indicated the presence of God, demonstrated that He had received the promised Holy Spirit from the Father and had here and now poured it out on His disciples.
So while his listeners had not themselves had the privilege that His disciples had had, of being witnesses to the resurrection, they had the next best thing, visible and aural evidence of His action in sending forth God's Spirit with power, which demonstrated His resurrection and present exalted position, as witnessed to by the words spoken by Him in their own tongues through His disciples. Let them see from this then that the reason for the empty tomb, of which they would all have heard, is that Jesus is risen, and that they themselves now have proof of it. The greater David has received what the first David could only wishfully hope for.