Basically supporters of the free silver movement thought that bimetallism would help the economy by causing inflation. This would help farmers and others who had too much debt.
Allowing bimetallism would have increased the amount of money that existed in the US. This would have happened because silver, and not just gold, would have been usable as money. (This was a time when the money had to be backed by gold, unlike today.)
If more money comes to exist in an economy, each dollar comes to be worth a bit less. If money comes to be worth less, those who are in debt benefit because the money they are paying back is worth less than the money they borrowed.
Supporters of free silver wanted to cause inflation to help farmers get out of debt more easily, thus (in their minds) helping the economy.
Farmers believed that a bi-metallic money supply would help them deal with the economic issues they faced. Farmers were suffering because prices for their crops were very low, and they had to borrow money at interest rates that were quite high. They believed that a money supply based on both gold and silver would help to alleviate their problems.
With a bi-metallic money supply, there would be more money in the economy. This would help raise the prices farmers got for their crops. When people have more money, they tend to spend it and demand more products. This helps to raise prices. Farmers believed this would lead to higher prices for their crops.
When there is an inflationary period, characterized by rising prices, those people who are debt benefit from the period of rising prices. When inflation occurs, the purchasing power of money drops. Thus, if the money supply increased by going to a two metal system when farmers repaid their debts it would be with dollars that could purchase less than the dollars could purchase when they originally got their loans. Plus, with more money in the economy, farmers might find it easier to get the money they needed to repay their debts.
Farmers believed a bi-metallic money supply would go a long way toward helping them solve their financial problems.