Why does over-whipped cream turn into butter?
Because whipping and churning are the same process taken to different points — and cream that has gone too far has not been ruined so much as converted into something else.
What cream is. An emulsion of fat droplets suspended in water, each droplet surrounded by a membrane of proteins and phospholipids that keeps it separate from its neighbours. Double cream is roughly 48% fat; single cream around 18%, which is why it will not whip at all.
What whipping does. Beating incorporates air and, crucially, damages the membranes around the fat globules. Partially exposed fat causes the globules to stick to one another, forming a network that surrounds and stabilises the air bubbles. That network is what makes whipped cream hold its shape.
Why it must be cold. At low temperature the fat is partly crystalline — solid enough to form a rigid network. Warm cream has fat that is too soft to hold structure, which is why warm cream will not whip and why chilling the bowl and whisk genuinely helps.
What happens if you keep going. More membranes break, more fat clumps together, and the clumps grow. Eventually the fat network collapses out of suspension entirely, squeezing out the water phase. The cream goes grainy, then separates into solid yellow butter and thin liquid buttermilk.
This is exactly how butter has always been made — churning is simply sustained agitation of cream past the point where whipped cream exists.
Can it be rescued? If it has only just turned grainy, gently folding in a little unwhipped cold cream by hand can bring it back. Once it has separated, it cannot — but you have butter, which is not a bad outcome. Drain the liquid, rinse the solids in cold water until the water runs clear, and knead out remaining buttermilk, which is what causes it to go rancid quickly.
What helps avoid it: cold everything; stop early, since cream continues to stiffen as you spoon and pipe it; and finish the last few strokes by hand, where the change is easy to feel.