The public has knowledge of the above because the National Football League files a Form 990 (Return of Organization Exempt from Income Tax) every year, just like Harvard University, the New York Public Library, and other tax-exempt organizations.
Internal Revenue Code section 501(c)(6) provides that tax-exempt organizations include "Business leagues, chambers of commerce, real-estate boards, boards of trade, or professional football leagues (whether or not administering a pension fund for football players), not organized for profit and no part of the net earnings of which inures to the benefit of any private shareholder or individual."
It should be noted that amateur football leagues are not covered by the tax-exemption, nor do associations for other sports, but they may qualify for the general tax exemption for charitable and educational organizations under section 501(c)(3).
An individual cannot claim a payment to the NFL as a tax-deductible charitable contribution, which are governed by different rules in section 170, but most payments by the teams to the NFL are deductible as business expenses.
The NFL obtained tax-exempt status from the IRS in 1942 (the original IRS filing has been lost), but section 501(c)(6) was modified to specifically include all professional football leagues in 1966 by Public Law 89-800 ("An act to suspend the investment credit and the allowance of accelerated depreciation in the case of certain real property"). After the House and the Senate both passed a bill concerning the investment credit and accelerated depreciation, the Conference Committee decided to add an extra provision for professional football leagues. The Conference Committee was concerned that the NFL's pension plan would otherwise be considered as benefiting a private individual.
The Conference Committee also helpfully amended the antitrust statute to provide that antitrust laws would not apply to the 1966 merger of the NFL and the AFL, though they might apply to future mergers in football and other professional sports.
Tax-exempt organizations are generally subject to federal income tax for conducting commercial activities. The Conference Committee tried to add an exception for the NFL's "income derived from promoting or sponsoring any professional football game if such promotion or sponsorship does not occur more than four times during the taxable year with respect to any team," but that proved even too much for the House and it was not included in the final legislation.