I don't know where I picked up that particular factoid, but if you look at the trend lines in [1] you can see the improvement claimed.

In [2] there is a slightly different claim, "A modern 5-day forecast is as accurate as a 1-day forecast was in 1980, and useful forecasts now reach 9 to 10 days into the future."

Chart 3.2 of [3] shows this; by 2001 the 5th day forecast improved to be as good as the 3rd day of 1980, establishing the trend line.

Googling about yields a few other studies and articles in a similar vein.

It is important to note that forecast improvement is not linear in effort. It takes more complete and accurate sensor data and far more computation to extend the forecast on the out days due to the chaotic nature of the mechanisms modeled.

[1] https://rmets.onlinelibrary.wiley.com/doi/full/10.1002/qj.25...

[2] https://science.sciencemag.org/content/363/6425/342

[3] https://www.nap.edu/read/10658/chapter/5#26

This was on here a while ago: https://news.ycombinator.com/item?id=19765700 and says that "[m]odern 72-hour predictions of hurricane tracks are more accurate than 24-hour forecasts were 40 years ago"

edit: I see that neuronexmachina found the same article. It's a good read if you want an overview of how weather prediction has changed.