How do illegal aliens file their taxes? Why does the U.S government let them do this?

If they file, and some do, then they apply for an ITIN. The ITIN is a Tax ID which is issued to a tax payer who does not qualify for a social security number. You file a form W-7 with the IRS along with the supporting documentation and then the number is issued.

Obviously many do not file and some file using fraudulent information but there is a method to pay for those who wish to comply with at least that portion of US law.

The IRS establishes a process to file for two reasons. First, it means revenue comes into the government. Second, the IRS is charged with collected revenue and does not have authority with regard to immigration matters. The separation of powers is often frustrating for Americans but it is part of our constitutional system. An agency of the government only has those powers granted to them in the enabling legislation. In similar fashion the IRS does not have authority to make arrest for those who report illegal income so long as they pay tax on that income. This does not mean that filing taxes with disclosure of illegal information will not result in arrest it just won't be through the IRS, it's someone else's job.

How would the IRS know whether a tax filer is legally able to work or live in the U.S.?

Tax forms do not ask that question, nor do they require the tax filer to prove it.

Instead, the government puts the burden on employers, who are required to verify that every employee is legally able to work in the U.S. (Immigration Reform and Control Act of 1986).

A better question is: Why doesn't the U.S. government check up on employers' compliance with the IRCA more frequently?

Illegals don't file taxes. They get phoney social security cards and claim the maximum number of exemptions so the minimum is withheld.

They don't file. Those who are using phony SS cards are probably losing the money (refund) that they would have been entitled to had they been legal.

Most don't file.