How did mining contribute to the development of the west?
It created a resource demand that increased settlement.
Other than pure adventurousness, nobody would have any reason to want to settle on the West. It's risky, uncolonized, there aren't many natural resources everywhere, and it's a huge costly effort. Mining changed that. Since pioneers/explorers started finding gold, silver, iron, etc. in the Rockies and other mountain ranges on the Western frontier, they realized the economic possibilities.
Soon, families were moving out to mine for gold, companies started building railroads to transport people there and get the metals and resources back to the Eastern factories, and homesteading became an increasingly lucrative prospect (homesteading was basically buying a large piece of land for the government at a very low cost and promising to do something with it, e.g. mining, and it aimed to get more settlers to colonize the frontier).