Here is a 3-paragraph summary for mathopen.com:
The Classification of Finite Simple Groups (CFSG) is one of the greatest collective achievements in the history of mathematics. Spanning decades of collaborative work by hundreds of mathematicians, the CFSG identifies every finite simple group, which are the fundamental building blocks of group theory, much like prime numbers are the building blocks of arithmetic. Mathematician Daniel Gorenstein declared the proof essentially complete roughly 45 years ago, though some loose ends remained at the time.
Those loose ends turned out to be significant. The full proof spans tens of thousands of pages across hundreds of research papers, making it nearly impossible for any single person to verify in its entirety. This raised genuine questions about whether the proof was truly airtight, and mathematicians have spent years working to fill gaps, simplify arguments, and organize the sprawling body of work into a more coherent and checkable form.
Now, that verification effort appears to have reached a meaningful milestone, with the proof being checked and confirmed. This is a big deal not just for group theory, but for mathematics as a whole, since CFSG underpins a vast number of results across many areas of the field. Having greater confidence in its correctness strengthens the foundation on which a huge amount of modern mathematics rests.