Briefly put, if you had travel expenses that you incurred as part of the work you did for the 1099-MISC and the reimbursement for those expenses were included in the total amount reported on the 1099-MISC (probably box 7), then you offset that "income" by listing them also as expenses on Schedule C.
When you have a 1099-MISC with an amount in box 7, then you in essence have a "business" and have to file a Schedule C, on which you can list all reasonable business expenses (with some limits).
In the Premier version of the TurboTax software, go to the Federal Taxes tab, then click on Wages & Income, then go down and click on Self-Employment. Walk through the interview, first answering questions about income (enter your 1099-MISC here), then answer the questions about expenses.
As you can see, this makes the income and deduction of the travel expenses a "wash" so that you don't pay any income tax on that amount.
Make sense?