Do I need to register with the State of Maine to pay employment taxes ONLY?
A: From the way you describe the work being performed, you would only be obligated for Maine employment/payroll taxes (unemployment, workers comp, Maine state income tax withholding), because although you could be considered to be engaged in business in Maine, you are not generating any revenue from resources located in Maine.
Of course, you could get audited, and the tax authorities may challenge your assertions about the conduct of your business activities. But, based on how you've described things, you would only be liable for employment taxes and withholding for the employee located in Maine.
Are there any changes to how I file my tax returns because of having out of state employees?
A: I don't believe that under your proposed scenario, that you would have any filing obligations in Maine, because you are an single-member LLC. You would only owe taxes on your income generated from Maine, in which case you would have to file a Maine nonresident personal income tax return. Your payroll costs would all be deducted on your PA state tax return (and on your federal Form 1040, Schedule C).
Note: there are outsourcing companies throughout the USA that handle administrative employees and the related withholding issues, so that the employer only need cut a check to the outsourcing firm, and the firm handles the employee financial relationship. You may want to google around for "outsourcing" and see if it would be more cost effective to let some other business handle the employee issues, rather than spend time trying to handle payroll matters yourself. Payroll wage and hour issues can be very complicated, are prone to changes in law on a frequent basis, and the fines for messing up are onerous.
Hope this helps.