It is very helpful to understand that Free File is not actually "by" the US Internal Revenue Service (IRS). The IRS does indeed offer access to the program through their website, but Free File is actually a public-private partnership program operated and maintained by the Free File Alliance. Who is the Free File Alliance? Well, according to their members list: 1040NOW Corp., Drake Enterprises, ezTaxReturn.com, FileYourTaxes, Free Tax Returns, H&R Block, Intuit, Jackson Hewitt, Liberty Tax, OnLine Taxes, TaxACT, TaxHawk, and TaxSlayer.
Why the income restriction? Well, that's part of the deal the IRS struck - the program is "dedicated to helping 70 percent of American taxpayers prepare and e-file their federal tax returns". Technically the member companies are offering their own software to handle tax preparation, and the rule is that 70% of American's must 'qualify' for at least one product, so this adjusted gross income limit changes periodically so that 70% of the population can use it.
Why restrict it at all? This was part of the give and take involved in negotiation with the businesses involved. If the program was "everyone files for free", then it is presumed that many reputable businesses that make the program valuable would choose not to continue to participate. In other words, they want to be able to not give away their services for free to customers who are - at least by income definition - more than capable of paying them. The IRS has said it does not want to be in the tax prep software business, so they are not offering their own free software to do the job that private companies would otherwise charge for.
However, there are other restrictions to being in the program - like the fact that no business in the program can offer "refund anticipation loans", offer commercial services more than a certain amount of times (so they can't hound you to upgrade), and so on. Some businesses were making a killing off these, though they are pretty much solely developed to be predatory on people with the lowest incomes (and education levels, and IQ, and with cognitive disabilities, and basically anyone they could sucker into paying what were effectively absurd rates for short term loans along with inflated filing/preparation fees).
Finally, Free File was partly developed as an initiative to increase the amount of digitally filed taxes and reduce the paper-based burdens of accepting and processing turns. In other words: to cut government costs, not to be a government welfare program. Even if it were, one can generally obtain commercial software for $30-$100, so the benefit to those above gross income levels is pretty minor; yearly costs to file taxes with such software for those payers would be less than 0.001% of their yearly expenses. Compared to the benefits obtainable by households living below the poverty line, fighting to cover an extra 5-30% of the population at the potential expense of having the whole program be a failure probably seemed like a more than worthwhile trade-off.