Why does my tax due go up when I include long term capital gains, even though I'm still in the 15% bracket, where long term capital gains are supposed to be taxed at 0%?

While your long-term capital gains may be eligible for 0% rates, it is still income itself, potentially impacting certain deductions and tax credits, and the taxation of Social Security. 

Since you are using the downloaded program switch to the FORMS mode and look at the QDCGT worksheet to see just how the tax is being calculated.  Also review your form 1040 before & after entering the stock sale to see just what is being effected.

When you enter one taxable transaction, you can't just watch the monitor.  You increased your overall adjusted gross income and with that come many other changes in your return, not just the incremental tax on the one transaction. Like for example it increased your AGI and that would decrease some deductions if you itemized on Schedule A.  And by increasing your AGI it might reduce some credits you were getting like EIC.

Thanks everyone...since I use the Standard Deduction, my tax increase was due entirely to having more of my SS eligible to be taxed because my "combined income" went up.

Bingo.