How did capitalism develop and secure its place in America's economic and legal systems?

Capitalism is the only economic system that is consistent with Natural Law.

Under the principles of Natural Law on which the US was founded, the "creator" whatever you believe that to be, endowed humans with 3 rights. Those are life, liberty, and private property. Life, is self explanatory. Liberty is the freedom to apply your life to the world around you for your own benefit or in the service of others. Private property is the compensation deemed adequate by the individual exchanging their life and liberty through labor.

Free people created governments for the sole purpose of protecting these rights. Government simply exists to make sure no one violates the rights of others. Government cannot violate the property rights of one individual without violating their life and liberty because the property they accumulate was exchanged for a part of their life and liberty. Taking from one to give to another or nationalizing wealth violates the very principles on which the US was founded.

Capitalism is the free exchange of goods and services between individuals or groups of individuals in a free society absent government intervention of any kind beyond what is absolutely necessary to enforce agreements between those governed.