We find here a strong effort by Peter to persuade the people to repent and be saved. As it happened, a lame man was just healed through Peter and John’s ministry (3:1-10)—for they had the gift of healing. And when the people saw that the man was healed, they ran after them as if they had supernatural power. But Peter took advantage of this opportunity to witness to them; and he was telling them that Jesus whom they denied was the one who really healed the lamb man. That is, faith in Him gave the man perfect health (v. 16).
In addition to preaching salvation to the people, Peter was also prophesying that God would send Jesus Christ back to the earth in order to bring a time of refreshing and restoration to the earth. Some think this is referring to the millennial kingdom. I think so too, but I think it also points further down the road to eternity, to the new heaven and earth, where all things are new (Rev. 21:1-5) Here then is another reference to the second phase of His coming, the coming that precedes the millennial kingdom.
“Repent therefore and return, that your sins may be wiped away, in order that times of refreshing may come from the presence of the Lord; 20 and that He may send Jesus, the Christ appointed for you, 21 whom heaven must receive until the period of restoration of all things about which God spoke by the mouth of His holy prophets from ancient time. NASB