A number of things: first was the person bold enough to stand before others and make tough, unpopular proclamations.  A true prophet usually had a negative message to deliver. False prophets usually said good things to make the people happy and be liked by them.

Then the officials and all the people said to the priests and the prophets, “This man should not be sentenced to death! He has spoken to us in the name of the Lord our God.” Jeremiah 26:16

            Second, were they bold enough to speak saying something "in the name of the Lord," or, declaring God’s authority.  If they were wrong, they could be stoned to death.  You’d better be willing to die if you prophesy.  Many were punished (like Jeremiah) and bounced right back with more bad news.  That should say something.

            Third, the people asked if the prophet’s prophecy came true.  In Hezekiah’s time, they repented for listening to a false prophet.

“Did Hezekiah king of Judah or anyone else in Judah put him to death? Did not Hezekiah fear the Lord and seek his favor? And did not the Lord relent, so that he did not bring the disaster he pronounced against them? We are about to bring a terrible disaster on ourselves!” Jeremiah 26:19

           The things Jeremiah, Isaiah, Jonah, Ezekiel said came true while the good things false prophets said did not.

           Fourth, the prophet spoke with authority and the people recognized it.  Jesus escaped death a number of times when the leaders called Him a false prophet.  However, they recognized over and over that Jesus spoke with authority.  John the Baptist too.

King Jehoiakim, however, sent Elnathan son of Akbor to Egypt, along with some other men. They brought Uriah out of Egypt and took him to King Jehoiakim, who had him struck down with a sword and his body thrown into the burial place of the common people. Jeremiah 26:22-23

            Killing good prophets, like Uriah, was a sure way to get God mad and bring His wrath on the people.  This happened after the Jews killed Uriah and especially Jesus (darkness filled the land).  There was a responsibility on the people to make sure the prophet was true and listen first before they responded with death.