What Does The Ending Of Joker REALLY Mean?
2. Arthur Gets What He Always Wanted
When it comes down to it, Arthur's reasons for doing what he did come down to him no longer being willing to be constantly beaten by people in power for their own pleasure - an allegory for his abuse as a child. He kills Murray because he exploited and mocked him, he killed Randall because he betrayed him, he was - effectively - doling out his own brand of justice.
But really, there's another motivation here that is more subtle. Early on Arthur talks about feeling best when he was in hospital and when he actually visits Arkham, he talks with the admin clerk and confesses his crimes when the clerk tells him why people find themselves locked up there. Arthur wants to be in Arkham and you can see it in all of his behaviour - not just in his crime, but in his very posture.
He walks around with his shoulders hunched and pulled across his body as if he's bound and at one point he writes a joker about a straitjacket being too tight. His whole demeanour and posture is designed to reflect someone who was comfortable wearing a straitjacket. And as further proof of that, when Arthur is most distraught, he empties his fridge and climbs in, creating himself a confined cell.
Arthur was locked away for his own good, but in a way it's where he felt most comfortable and most safe (which you would if you were sent back to the same mother who allowed your abuse). When he ends up there at the end, it's a realisation of his real dream.