A curling stone leaves the hack rotating slowly around its own vertical axis (the "handle"). The running band under the stone contacts the pebbled ice asymmetrically as it spins — the friction pushing back on the leading edge differs slightly from the trailing edge — and that tiny sideways force accumulates over the whole 30+ meter slide into the visible curl, always toward the side the stone is turning.
A slower stone spends more time on the ice per meter travelled, so the same asymmetric force has longer to act and the stone curls more; a hard, fast throw barely curls at all. Ice slipperiness sets the base friction coefficient: colder, more heavily pebbled ice runs faster (lower μ) and curls more per throw.
Sweeping briefly melts a microscopic film of water in front of the stone, cutting friction — this both carries the stone farther and, because it shortens the time spent scraping the ice per meter, straightens its path by reducing the curl. Sweep hard while the stone slides (drag across the ice ahead of it) to feel both effects at once.