ScienceClic on youtube has some really great videos, the first half of the string theory video [1] does a good job framing the conflict at a high level, string theory being one attempt to resolve the problem.

It doesn't go into math but mentions that in the standard model, interactions are local, which for example means something like photon emission is instantaneous. When applied to gravity/gravitons, you get infinities in the math. This sort of makes sense to me, but I have yet to find anything that does an understandable job explaining how/why exactly the math breaks down.

There's a series on the maths of general relativity [2] which I haven't completed but I'm hoping will provide a bit more background to understand the math.

Also, the article mentions that the new theory was motivated by solving the black hole information paradox [3], which there's a good video [4] about as well

[1] https://www.youtube.com/watch?v=n7cOlBxtKSo

[2] https://www.youtube.com/watch?v=xodtfM1r9FA&list=PLu7cY2CPiR...

[3] https://en.wikipedia.org/wiki/Black_hole_information_paradox

[4] https://www.youtube.com/watch?v=isezfMo8kWQ