Great question! Simply put, there's a strong demand for silver. All new silver products have to come from somewhere. Whether it's for industrial use, jewelry, or coins, the silver coming from mining doesn't always meet the immediate demand. Silver is very difficult to mine by itself, and most mined silver is actually byproduct from other mining.
If a coin dealer has twenty bags of junk silver, he'll make a bigger profit selling the silver to a refiner than he would selling the coins individually---and that's only because it would theoretically take an
extremely long time to sell each individual coin at a higher profit than for what the refiner would pay him all at once upfront. In other words, demand primarily exists for high-grade coins. Collectors won't pay a premium for common, worn down junk. Coin dealers are simply the middle men who accumulate large quantities of silver, which is the only profitable way to sell to a refiner. Collectors don't have enough silver to sell directly to a refiner (except maybe LJ.

), so we must rely upon the coin dealer when we sell. The refiner pays the dealer more money for the silver than what the dealer pays us.
And from what I can gather, refiners don't like refining silver as much as other precious metals, because, when all is said and done, there's less likely to be valuable "precious metal sludge" at the bottom of the vat. Raw, unrefined platinum, palladium, and gold is variously alloyed with platinum, palladium, and gold, but after the metals are purified and separated, the refiners get to keep whatever is left over. When refining silver, you probably aren't going to have much platinum sludge left over.
After the silver has been purified, it is then either sold back for a fee to the original seller if he/she wants it, or it is bought outright (most common) and then, when quantities allow, it is poured into what is known as a Good Delivery Bar and then sold to the highest contracted bidder. A gold Good Delivery bar must weigh between 350 and 430 ounces, and it must be assayed with a purity of at least 999.5. I don't remember the exact weight specifications for silver Good Delivery Bars, but it's approximately 1000 ounces per bar. One Good Delivery Bar is the minumum accepted amount of gold or silver deliverable against a metals contract. Spot prices, as determined in London every day, are based upon the buying and selling of Good Delivery Bars.