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The Republicans by the 1890s were supporters of "banking" interests, and "rich people" in general.
Over time, more and more silver was discovered relative to gold. At first, it required 14 ounces of silver to equal one ounce of gold; later it required 16 ounces. (In modern times, the silver-gold parity has ranged between 50 to 100 ounces of silver for one ounce of gold).
"Bankers liked to denominate their loans in gold. So if they lent you one ounce of gold (worth 14 ounces of silver in say, 1890), by 1896 that ounce of gold was worth 16 ounces of silver. But the borrower (e.g a farmer) got only 14 ounces of silver, earlier on, and now had to repay 16 ounces of silver, plus interest.
"Bi-metallism" would mean that (poor) farmers (who mainly transacted in silver) would have the choice of repaying 14 ounces of silver or one ounce of gold for each ounce of gold they borrowed; a "gold standard" would force those farmers to pay 16 ounces of silver.That's why the farmers would want bimetallism,and the bankers wouldn't. The bankers won by electing Republican William McKinley over Democrat William Jennings Bryant President in the 1896 election.
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The essence of the question is that the use of silver, in the exchange ratios desired, would result in inflation and lower the cost of paying off debts as it was really a devaluation in terms of gold. It was a debt cancellation move that would benefit debt-stricken farmers at the expense of bankers and other elites. A good general discussion of debt cancellation is the book Debt: The first 5000 years.