"- you c - can't sack me! I've b - been here sixteen years! H - Hogwarts is m - my h - home!"
-- Sybill Trelawney (OP26)
Umbridge uses the power given to her by Educational Decree Number Twenty-three to sack Trelawney from her post as Divination teacher. Crowds of students gather as she attempts to throw a wailing Trelawney out of the castle. Dumbledore intervenes as the power to decide who may live in the castle still resides with him. He then infuriates Umbridge further by introducing the replacement Divination teacher he has found: Firenze.
Trelawney's scream interrupts Harry's Occlumency lesson, which was taking place 'a couple of weeks' after his dream about Rookwood.
On first reading it seems that Dumbledore saves Trelawney from being ejected from the castle purely out of kindness. We find out later on, however, that Dumbledore has an ulterior motive for keeping Trelawney at Hogwarts - she was the one who made the prophecy regarding Harry and Voldemort.
McGonagall's kindness towards Trelawney is particularly touching here as we know that she does not usually have much patience with her.