Why / How Hagrid was framed and expelled

'They think, up at the school, that Hagrid's been setting a- a- something on students. They've taken him to Azkaban.'
Aragog clicked his pincers furiously, and all around the hollow the sound was echoed by the crowd of spiders; it was like applause, except applause didn't usually make Harry feel sick with fear.

'But that was years ago,' said Aragog fretfully. 'Years and years ago. I remember it well. That's why they made him leave the school. They believed that I was the monster that dwells in what they call the Chamber of Secrets. They thought that Hagrid had opened the Chamber and set me free.'
'And you... you didn't come from the Chamber of Secrets?' said Harry , who could feel cold sweat h=on his forehead.
'I!' said Aragog, clicking angrily. 'I was not born in the castle. I come from a distant land. A traveler gave me to Hagrid when I was an egg. Hagrid was only a boy, but he cared for me, hidden in a cupboard in the castle, feeding me on scraps from the table. Hagrid is my good friend, and a good man. When I was discovered, and blamed for the death of a girl, he protected me. I have lived here in the Forest ever since, where Hagrid still visits me. He even found me a wife, Mosag, and you see how our family has grown, all through Hagrid's goodness...'

ARAGOG - Pg 293
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So I decided to show you my famous capture of that great oaf, Hagrid, to gain your trust.'
You framed him, didn't you? I thought you made mistake, but - '
'It was my word against Hagrid's, Harry. Well, you can imagine how it looked to old Armando Dippet. On the one hand, Tom Riddle, poor but brilliant, parent-less but so brave, school Prefect, model student; on the other hand, big, blundering Hagrid, in trouble every other week, trying to raise werewolf cubs under his bed, sneaking off to the Forbidden Forest to wrestle trolls. But I admit, even I was surprised how well the plan worked. I thought someone must realise that Hagrid couldn't possibly be the heir of Slytherin. It had taken me five whole years to find out everything I could about the Chamber of Secrets and discover the secret entrance... as though Hagrid had the brains, or the power!

THE HEIR OF SLYTHERINE - Pg 328

A girl was killed and the school was about to be closed for good, which Tom Riddle did not wanted to happen. So Tom Riddle framed Hagrid gave them 'A culprit'; and the case was closed for Ministry of Magic. Dumbledore must have started to suspect Tom Riddle from then on knowing full-well of Hagrid's innocence. Hagrid was officially acquitted only after the events in Chamber of Secrets, when it was proven that Tom Riddle had been the Heir of Slytherin.

Hagrid wasn't a fully qualified wizard; he had been expelled from Hogwarts in his third year, for a crime he had not committed. It had been Harry, Ron and Hermione who had cleared Hagrid's name last year. - THE DEMENTOR, Prisoner of Azkaban Pg 99