Proof-based is an approach for teaching math. I'm having trouble coming up with a definition (math pedagogy isn't my area of expertise by a long shot!). Basically it involves teaching/learning through proving mathematical statements.
So instead of a question like "Find the area of the triangle" you would have a question like "Demonstrate that the two triangles are similar". So it's not enough to know/be able to find the answer, but you also have to explain your logical path to that answer - you prove it.
Like Seraphia, it took me a while to get the point of it. But in a way its a set up for more advanced mathematics. I only asked because if your son is struggling with proofs, I suspect a little bit of tutoring or after hours help would go a long way in helping him understand and do well in geometry. Once the concept of "proofs" gets solidified, a lot of the other geometry concepts finally click into place. Based on my experiences tutoring high school students, it seems many classes don't spend enough time making sure students understand proofs, and then the students just give up and assume they aren't good at geometry. It would be akin to mistakenly thinking a student was bad a literary analysis when in actuality the student never learned how to read properly.