10% is in quarter-on-quarter annualized terms. In QoQ it's -2.6%, following a -1,0% in the last quarter of 1957. And the recovery is 10.0% in annualized terms as well. The National Bureau of Economic Research (NBER) dates a recession around that time, starting in August 1957 and ending in April 1958. With 8 months, this is actually one of the shortest recessions in the post-WW2 era in the US (disregard the 6 month recession in 1980, most economists I know say it should actually be put together with the next recession, that started just 12 months later in 1981).
http://www.nber.org/cycles.html
The story behind this economic cycle is not that different from others in post-WW2 American history. To sum up, inflation increased, so the Federal Reserve increased interest rates, this caused the economy to contract. The Fed then decreased interest rates and the government used fiscal policy to stimulate the economy, following the Keynesian economic recommendation of the time. This resulted in a very quick recovery.
If you look at the Federal Reserve data on interest rates
You can see that there was an increase in the Fed discount rate (the last column) from 1.5 in 1955 to 3 in 1956. And then in the last quarter of 1957 it gets raised even more, from 3 to 3.5. The 1 year market interest rate increases even more, to 4 at the third quarter of 1957. Interest rate increases affect the economy with a lag (that varies from country to country and from time to time, but most economic studies put it at 6 months to 2 years). So this increase in interest rates caused the economy to contract a year later, affecting housing prices, the construction sector and auto sales. Regarding auto sales, Ford released a terrible car in that year, with terrible sales and net losses to the company, and this also increased the size of the recession, but it was a temporary effect that had no bearings on the economy overall and could be responsible for the large numbers that misrepresent it.