How did the New Deal try to solve the problems of the Great Depression?

There were many components to the New Deal that we cannot cover in this short of a space, but the overall strategy employed by FDR was to put into action the economic theories of John Maynard Keynes, who posited that during a great economic recession, private money would flee the market and the government, in order to turn the tide, needed to inject lots of money quickly into the economy, to stimulate growth and reverse the vicious cycle that results from loss of confidence in the market, tightening credit, business failures and job losses. The way the New Deal injected money into the economy was through what is called "Deficit Spending," a process by which the government borrows money and prints excess currency to pay for programs that put people back to work.

The New Deal can be divided into two main components: stimulative action and regulation. Stimulative action included the Social Security Act, which gave widows and orphans direct payments to help them afford basic necessities. This also included unemployment assistance, direct cash payments to unemployed workers to help them pay for basic necessities and in turn, support the businesses that provided those services.

Other components of the stimulative side of the New Deal include the various works projects, like the Works Project Administration, Civilian Conservation Corps, Agriculture Adjustment Bureau, and the Tennessee Valley Authority, among others. All of these programs put unemployed men (and women) to work on large infrastructure projects, thereby improving and expanding the United States' roads, bridges, waterways, farms, irrigation systems, national parks and cities, and creating a virtuous economic cycle of job creation and consumer spending, which in turn prompted more job creation and more consumer spending. 

The regulation component of the New Deal was just as important, because it aimed to reform the failed mechanisms that allowed the financial collapse to occur in the first place. The regulatory programs of the New Deal included (but were not limited to) the Federal Depositors Insurance Corporation (FDIC), to insure depositor's money, the Glass Steagall Act, to prevent savings and loans banks from gambling their depositors money on Wall Street, the creation of the Securities and Exchange Commission (SEC), to police and monitor the trading and marketing of securities on Wall Street, so that investment bankers couldn't sell fraudulent securities and could not lie to their counterparts about the risks  involved in the investments they sold. The regulations also included a Bank Holiday at the very start of the New Deal, to stem the flow of bank runs and return confidence to the financial system, before other regulation could be enacted. The National Industrial Recovery Act (NIRA) aimed to regulate and improve business practices and labor conditions.

All in all, the New Deal was the single largest and most effective government intervention into the economy in this country's history. It succeeded in digging the country out of a long and deep economic depression by allowing the federal government to act as the "lender of last resort," when private citizens and private institutions were unable and/or unwilling to step in.