Notice verse 12? "Elijah has come already" because He was speaking about John the Baptist. How did John the Baptist "restore all things"? It certainly can't be some restoration of the Gospel, as the proclamation of the Gospel didn't begin until after John was arrested,
"Now after John was arrested, Jesus came into Galilee, proclaiming the gospel of God, and saying, 'The time is fulfilled, and the kingdom of God is at hand; repent and believe the good news.'" - Mark 1:14-15
But what did John do?
"Do not be afraid, Zechariah, for your prayer has been heard, and your wife Elizabeth will bear you a son, and you shall call his name John. And you will have joy and gladness, and many will rejoice at his birth, for he will be great before the Lord. And he must not drink wine or strong drink, and he will be filled with the Holy Spirit, even from his mother's womb. And he will turn many of the children of Israel to the Lord their God, and he will go before him in the spirit and power of Elijah, to turn the hearts of the fathers to the children, and the disobedient to the wisdom of the just, to make ready for the Lord a people prepared." - Luke 1:13-17
This, as it was written:
"Behold, I will send you Elijah the prophet before the great and awesome day of the Lord comes. And he will turn the hearts of fathers to their children and the hearts of children to their fathers, lest I come and strike the land with a decree of utter destruction." - Malachi 4:5-6
"A voice cries:
'In the wilderness prepare the way of the Lord;
make straight in the desert a highway for our God.
Every valley shall be lifted up,
and every mountain and hill be made low;
the uneven ground shall become level,
and the rough places a plain.
And the glory of the Lord shall be revealed,
and all flesh shall see it together,
for the mouth of the Lord has spoken.'" - Isaiah 40:3-5
"Then Jesus came from Galilee to the Jordan to John, to be baptized by him. John would have prevented him, saying, 'I need to be baptized by you, and do you come to me?' But Jesus answered him, 'Let it be so now, for thus it is fitting for us to fulfill all righteousness.' Then he consented. And when Jesus was baptized, immediately he went up from the water, and behold, the heavens were opened to him, and he saw the Spirit of God descending like a dove and coming to rest on him; and behold, a voice from heaven said, 'This is my beloved Son, with whom I am well pleased.'" - Matthew 3:13-17
With this has been fulfilled all which was hoped for and promised, the Messiah has come. And so the work of the Forerunner was accomplished, as he himself says,
"John answered, 'A person cannot receive even one thing unless it is given him from heaven. You yourselves bear me witness, that I said, "I am not the Christ, but I have been sent before him." The one who has the bride is the bridegroom. The friend of the bridegroom, who stands and hears him, rejoices greatly at the bridegroom's voice. Therefore this joy of mine is now complete. He must increase, but I must decrease.'" - John 3:27-30
In this way Elijah came and restored all things, even as the ancient prophets had foretold.
-CryptoLutheran