The "pay twice" means once on your tax return, and again whenever you withdraw the money (usually in retirement).
Let's think about an IRA first. With a traditional IRA, you get a tax deduction for your contributions. Since the money is not taxed on deposit, all the funds are taxed on withdrawal. But suppose you deposited $10,000 without taking the tax deduction (after-tax contribution; there are reasons you might do this but they are uncommon). Now you are ready to retire and you have $100,000 in your IRA. If you withdraw it all, you pay tax on only $90,000, because $10,000 was already taxed.
With a 401k, that is not allowed. If you make an excess contribution, the excess is added back to your taxable income, so it is an after-tax contribution. But you don't track the after-tax contributions separately, and all your withdrawals are taxed when you retire. So that amount of money was taxed on your income in the year you earned it, and will be taxed again whenever you withdraw it.
