How can my DNA test say 'less than 1% Neanderthal' if Neanderthals went extinct 40,000 years ago?

Because Neanderthals didn't fully go extinct. They interbred with modern humans, and a small percentage of their DNA is still walking around inside everyone of non-African descent.

When anatomically modern Homo sapiens migrated out of Africa starting around 60,000 years ago, they encountered Neanderthals in the Middle East and Europe. The two groups interbred enough that today, every person of European, Asian, or Indigenous American descent carries between 1% and 4% Neanderthal DNA. (People of fully sub-Saharan African descent carry less, though recent research has shown that back-migration brought small amounts of Neanderthal DNA into some African populations too.)

The "less than 1%" in your report is the fraction of your specific 23 chromosome pairs that matches the Neanderthal reference genome. The fraction varies between individuals - siblings can have noticeably different Neanderthal percentages because of how DNA is shuffled in each generation.

What does Neanderthal DNA actually do in modern humans? Some of it shows up in immune function, some in hair and skin traits, some in pain sensitivity. Some of it is just background. The science of identifying Neanderthal contributions to modern health is genuinely interesting and still developing.

Adding a fun fact: Denisovan DNA is the other half of this story.

Denisovans are an extinct human cousin we only learned about in 2010, from a finger bone found in a Siberian cave. They also interbred with modern humans. People of East Asian and Melanesian descent carry small amounts of Denisovan DNA - up to 5% in some Papuan populations.

We have learned almost everything we know about Denisovans from their DNA inside us. There are still no confirmed Denisovan skeletons beyond a handful of bone fragments. They are, in a real sense, ghosts who are mostly visible through their descendants.