This is kind of correct, kind of incorrect.
If a basic rate taxpayer wants a charity to benefit by 100, they can give them 78 under gift aid, which enables the charity to reclaim 22 from the government. This is economically the same (ignoring admin costs) to giving a basic rate taxpayer relief on a donation of £100 @ 22%. Effectively relief is given "at source" by the payment from the government directly to the charity. Doing it this way avoids the need for basic rate taxpayers to make a claim to receive relief (a large number of UK employees, particularly basic rate taxpayers do not need to complete tax returns as all tax is collected through the PAYE system.) This is because the UK has very, very few deductions available from employment income (pension contributions being the main one).
A higher rate taxpayer who wants a charity to benefit by 100 does the same, but can then claim relief from higher rate tax on the income, meaning there tax bill is reduced by £18, so the net cost to them is £60. This is the same as giving tax relief on the whole donation at 40%. Part of the relief is "at source" as for basic rate payers, part directly on a claim in the tax return.