In episode 345 of Naruto Shippuden (オレは地獄に居る/I’m in hell), when Obito arrives on the battle field, he sees how Kakashi killed Rin. The wiki summarizes this scene as
On his way to the battle site, Obito has a vision of Rin through his empty eye-socket. He arrives just in time to see Kakashi run his Chidori through Rin’s chest. Rin’s death awakens both their Mangekyō Sharingan.
In the next scene we see that the Kiri-nin are clearly disappointed by the fact that Kakashi killed Rin, one of them says “Damn it! He did it!”, and another replies “After all the trouble we went through to get her!”
But I don’t understand why he had to kill her, why they couldn’t wait, e.g., for reinforcements to arrive. Is there some good reason why it was better, for her, or for the village, that she dies, rather than to be captured by the Kiri-nin?
When they were on a mission in the Hidden Mist, the Hidden Mist ninja captured Rin and made her into the 3 tails’ Jinchuuriki. Their purpose was for Rin to return to the village, where they’ll undo the seal, kill Rin and have the 3 tails rampage on the Hidden Leaf.
Knowing that, Rin jumped in front of Kakashi’s Chidori as he charged on a hidden Mist ninja to kill herself and foil their plans.
Obito came a second too late and only saw Rin getting impaled, but later on he too learned the whole story.
The Hidden Mist shinobi were manipulated by Madara. He was the one who set that plan up then told Obito so it could have awaken his power. Rin had a curse on her heart placed by Madara.