This article isn't perfectly scientifically accurate but it's better than most! For those interested, here's a better (detailed) explanation of the science:

Cloud seeding works best in supercooled liquid clouds, which start with barely any ice in. For ice to form in clouds, you need INP (ice nucleating particles), aerosols such as sea salt, or dust, for the ice to grow onto. INP can be pretty rare, depending on where you are in the world.

By adding silver iodide (an efficient INP) ice crystals can form. This means the cloud has both ice and liquid water in (a "mixed phase" cloud).

For multiple reasons, ice grows better and faster than liquid cloud droplets (the "Wegener Bergeron Findeisen effect" for one). Because there's only so much water in the cloud, these ice crystals then grow at the expense of water droplets in the air, allowing for big snowflakes to grow, but droplets evaporate.

This turns it from a cloud with many tiny droplets, into one with big heavy snowflakes, which fall out of the cloud. Before the snowflakes reach the ground, they melt, turning into rain.

TLDR; cloud seeding takes the water already in the cloud, and makes it precipitate slightly more efficiently, but only if you improve the balance of aerosols in the air just right