Behold, I will send you Elijah the prophet

Before the coming of the great and dreadful day of the LORD.
 And he will turn
The hearts of the fathers to the children,
And the hearts of the children to their fathers,
Lest I come and strike the earth with a curse.”
Malachi 4:5-6

That is how they knew he was coming again, they were told directly that he would come. This was said after Elijah's death. So Elijah was dead and they were told that he was going to come.
In my opinion, they were as anxious about Elijah coming as they were about God coming.
This obviously comes from a wrong view of things. They were excited about Elijah becaue they had heard and seen the miracles he performed through Jesus. But they did not see that the miricles were all done by Jesus through him. They didn't think about the wonder of the miracles God could do in His own body.

They asked him, "Then who are you? Are you Elijah?"

He said, "I am not."
"Are you the Prophet?"
He answered, "No."
John 1:21

That was the people questioning John. They asked him who are you and he said I am not the Lord. Then they asked about Elijah.

As John's disciples were leaving, Jesus began to speak to the crowd about John: "What did you go out into the desert to see? A reed swayed by the wind? If not, what did you go out to see? A man dressed in fine clothes? No, those who wear fine clothes are in kings' palaces. Then what did you go out to see? A prophet? Yes, I tell you, and more than a prophet. This is the one about whom it is written:


" 'I will send my messenger ahead of you, who will prepare your way before you.' I tell you the truth: Among those born of women there has not risen anyone greater than John the Baptist; yet he who is least in the kingdom of heaven is greater than he. From the days of John the Baptist until now, the kingdom of heaven has been forcefully advancing, and forceful men lay hold of it. For all the Prophets and the Law prophesied until John. And if you are willing to accept it, he is the Elijah who was to come.
Matthew 11:7-14

So John is the Elijah. He is not the man Elijah, but he plays Elijah's role in life. So Matthew 4: 5-6 was a right, he did come but not in the exact same form.
The life of Elijah is very interesting and confusing. But... that is what I think.
The prophesy has been fulfilled, and the people had a wrong view of Elijah, over God.