FDR used it as part of a plan to fight the Great Depression and get the US economy moving again. FDR needed to expand the money supply, more on why this would fight the depression below, he needed to print more money. But the US was on the Gold Standard; you couldn't just print more money, every dollar had to be backed by gold. So to fight deflation the US needed more gold. To get it, FDR artificially raised the price at which the US was buying gold with the Gold Reserve Act.
He also forced US citizens to sell their gold to the US, at a very good rate mind you. This had three effects. First, the US got more gold by forcing US citizens to sell to the government at a very good price. This wasn't the only way the US govt got gold, they got a lot from foreign investors.
Second, it prevented more gold, "hard currency", from leaving the US. As the US economy crumbled, US citizens would be looking to protect their money by investing in foreign countries; foreign countries who wanted gold. As gold left the country the money supply would shrink causing deflation and worsening the depression. To prevent this, US citizens owning or trading gold was outlawed a year prior to the Gold Reserve Act.
Third, it prevented private speculators exploiting the artificially raised price of gold. Since the Act nearly doubled the price of gold overnight, such an upheaval in the gold trade might have lead to hording and speculation. The US govt wanted to control the price of gold so it was assured people would sell to them, and in order to do that it had to effectively shut down the gold trade within the US. If the market price of gold in the US went over what the US govt was offering, people wouldn't sell to the govt and the whole plan would fall apart.
In short, because money was gold, this allowed the US govt to effectively control their own money supply.
One of the major factors in an economic depression is a shrinking money supply. As the economy falters, investors are less willing to invest their money, and people are less willing to spend it. If investors aren't investing, businesses can't raise capital to expand and hire people. If people aren't getting hired, they have less money to buy things. If people aren't buying, businesses can't keep paying their workers...
As there's less money to go around, supply shrinks and it becomes more valuable. While this is great if you have money, it's not so good if you want to run a business. A mild, predictable amount of inflation is good for a modern economy.
Rather than letting it sit around gaining value through deflation, with a mild inflation people need to make their money work to retain and gain value. They spend it, or they loan it out. This keeps money circulating, further increases the apparent money supply (ie. what's available for use rather than being kept under the mattress), and lets a capitalist economy run.
I don't want to get into a whole economic thesis here, the point is one of the ways you can fight deflation and economic depression is to literally print more money. But at the time the US, and most other countries, were on the Gold Standard. That meant every dollar was backed by a certain amount of gold. Not 100%, but enough that it would retain some value.
The Gold Standard limits the amount of control a government has over its monetary policy. If you're anti-government, this is a good thing as it prevents them from just printing more money to pay the bills and causing hyperinflation. If you're pro-government, this is a bad thing as it prevents them from managing the economy, leaving it in the hands of businesses, and preventing things like the Great Depression.
We're allowed to own gold today because we no longer use the Gold Standard or any material thing to back our money, known as representative money. Today's money is backed by our faith in the government to repay its debts, our faith they won't print too much money, and the value the markets say it has: fiat money. Governments have control over their fiscal policy, and private gold speculators have only a very mild effect.
For a general overview of all this, I highly recommend Extra History's series on The History Of Paper Money.