Why does going to war help the US economy?

No. It doesn't help. It used to be thought that way until Keynes showed that any massive increase in government expenses could stimulate an economy in depression. The War itself had a negative effect, but the increase in spending that accompanied that acted against the depression.
However, in non-deflationary periods (such as during Vietnam war), wars substantially worsened both inflation and GDP growth.

1. Wars boost inflation

Vietnam and the World Wars substantially pushed up the prices. Afghanistan/Iraq war did that too, until the 2008 recession overcompensated for that.

2. Wars Shatter Market returns

First, here are the DJIA (US Stock market) returns per decade along with the major wars that decade. Decade is a good unit to see trends and patterns. You can see clearly what happens to the market in the decades of the war.

1. 1900s +50.69%             Peacetime
2. 1910s +8.26%                WW-I
3. 1920s +131.73%        Peacetime
4. 1930s -39.64%               Depression and start of WW-II
5. 1940s +33.69%              World War II
6. 1950s +238.80%        Peacetime from 1953
7. 1960s +17.81%               Vietnam war (most of the decade)
8. 1970s +4.80%               Vietnam war (until 1975) and Middleast wars
9. 1980s +228.25%            Peacetime
10. 1990s +317.59%         Peacetime (but for 1991-92)
11. 2000s -8.34%                 Afghanistan/Iraq wars

Some of the best periods for the stock market were 1954-59, 1993-99 were the peacetimes. 

3. Wars push the Government into Debt

Here is US debt during war times. See the debt spiking substantially during each war and the decades it takes to stabilize the debt. 

Wars push down the markets and push up the debt. The spike of debt gives a temporary high that feels like GDP growth. World War II got US out of Great Depression, but the same could have been done by any major increase in government expenses. 

Here is what the Nobel winning Joseph Stiglitz concludes on this:

World War II is widely thought to have helped lift the global economy out of the Great Depression. But, at least since Keynes, we know how to stimulate the economy more effectively

America’s War-Torn Economy by Joseph E. Stiglitz