Kalen Delhaney has a couple of posts on monitoring or getting the age of your statistics (http://sqlblog.com/blogs/kalen_delaney/archive/2013/08/31/how-old-are-my-statistics.aspx).
The query analyzer engine uses the distribution statistics, including the data distribution of indexes and column data to estimate optimal queries. These are stored in the sys.sysindex table. If the statistics become stale and the distribution of the data changes, the estimated plan can be badly calculated and the query at risk of running poorly.
DBCC show_statistics and a number of DMV/DMFs as well as some queries can tell you the last time that the statistics were updated.