Humanity is new to Anya, so she is certainly trying to process how Buffy's friends are reacting to Joyce's passing. For example, she doesn't understand the emotional and psychological reasons behind Willow continuing to change her clothes, but she wants to know why. However, Anya is actually not that far off emotionally from everyone for once; they're all disoriented, upset, and trying to cope. Anya is scolded by Willow for asking questions about the body that are too direct, which is typical for Anya to do, but I believe she is trying to maintain her own sense of normalcy as well. With her speech come tears and sincere emotions about her anger over death, and how she finally knows what it's like to be hurt by death's ripple effect and the uncertainty it creates.
Anya's speech articulates what Buffy can't since she is in a state of shock. Additionally, the other characters express/perform some of the various stages of grief for Buffy:
1) Denial = Dawn doesn't believe Buffy when she tells her the news at school.
2) Anger = Xander punches the wall and blames Glory (the demon) and doctors.
3) Bargaining = Willow changes clothes several times to stall time before they go see Buffy at the morgue.
4) Depression = Anya's teary speech and sense hopelessness.
5) Acceptance = Tara talks with Buffy at the hospital about death. Buffy asks about Tara's mother's death, "Was it sudden?" to which Tara replies, "It always is." Tara also elaborates how death is different for everyone and it's never easy.
Buffy's friends literally help her cope by functioning as various stages of grief. As Xander says: "We'll help. That's what we do. We help Buffy."