Your SSN is needed to determine a number of things (legal residency, comparing 2014 income estimates to known 2012 data, etc), but most importantly it tracks any federal tax credits that you are given when/if you purchase a policy through the exchange. This way they can reconcile your actual income with your estimated income for the year that was used to determine your subsidy.
If you got too small of a subsidy (estimated income higher than actual income), you would receive the rest as part of your income tax return (when your actual income is now a concrete number). If you received too large of a subsidy (estimated income was lower than actual income) you repay the overage when you file your taxes. The actual rules are a bit more complex, but that captures the gist of it.