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Does the trust make a check to the IRS and to me for my cost?"
Yes, and then the trust can deduct the amount it paid you for the cost of the tax preparation software on its 1041.
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How is that handled next year when I'm asked to enter distributions from the trust?"
In the event a distribution is made (or deemed to be made per the trust instrument), then the distribution is passed through on the K-1 as taxable income to the beneficiary to the extent of distributable net income (DNI). The trust, of course, receives an income distribution deduction (IDD) which generally results in the trust having no income tax liability (i.e., tax liability is shifted to the beneficiary).
