Basically what the title says... What's happening to my peonies? Also, what can I do to help?

Radishgirl, not sure but it looks like a fungus coming up from the roots. Is the soil very wet. Hopefully, someone with knowledge can help.

Botrytis. Really bad. Usually it is recommended to dispose the plant and the soil around it. You have to let us know the plant history and may be we can come up with a reason why it happened.
Dispose it if it is not crazy expensive and valuable peony.

There is peony forum here and people there will give you more detailed advice.