The article was last updated in October 2021, but the data in the charts is current insofar as it's available.
Their methodology is posted on Github here: https://github.com/TheEconomist/covid-19-excess-deaths-track...
They get their data from the World Mortality Dataset (https://github.com/akarlinsky/world_mortality). They claim their modeled baselines fit a linear trend for year, to account for long-term increases or decreases in mortality, and a fixed effect for each week or month up to February 2020.
I haven't dug into the details of the methodology but it appears very sensible. Of course it's very dependent on overall mortality rates having been accurately reported to begin with.
If the Economist keeps this this chart going long enough, we should expect that in countries worst affected by covid, future mortality rates will fall below the expected baseline. The simple (if somewhat morbid) logic being that in those places the most vulnerable people will have already died.