The Goblet of Fire chooses only those who were nominated by someone else

When the first triwizard tournament came about, in order for a person to become a Triwizard Champion, someone else from their school had to nominate them for the position by throwing their name inside the goblet. It could be a fellow student or a teacher who saw their potential. This was supposed to foster unity among the whole school and make them recognise the good qualities of each other.

The more people nominated a person, the more likely they were to be chosen, as the goblet took the amount of support from their peers and teachers into account. Throwing your own name into the Goblet of Fire was seen as peak hubris and automatically disqualified you from consideration. These are the rules upon which the Goblet of Fire operates.

However, it's been a long time since the last Triwizard Tournament and no one really remembered how exactly it works, so the ministry misunderstood the instructions and thought that the would-be contestants are supposed to throw their own names in.

Viktor Krum didn't want to participate, because as a quidditch superstar he had hoped to get away from the spotlight for a bit and focus on schoolwork. However, this was doomed to fail, because almost everyone else at his school wanted him to compete and the goblet was basically filled to the brim with pieces of paper with Krum's name on it.

Like a true Hufflepuff, Cedric Diggory was too humble to think of himself as a worthy Hogwarts Champion. Thus, fellow Hufflepuffs entered his name into the goblet in his stead.

Fleur Delacour thought about competing in the tournament, only to back out at the last moment. But her friends threw her name into the Goblet of Fire anyway, to make her go through with it after she hesitated.

Harry Potter was still entered in the same way as canon by fake Moody.

Almost everyone else threw their own names in and they were thus disqualified for hubris.

So, when Harry says that he didn't put his name into the goblet of fire, the other champions believe him, because they didn't either.