The "Background and history" section of the following article gives a very high-level idea of Grothendieck's contribution:

https://en.wikipedia.org/wiki/Weil_conjectures

What is the modern path to study this area now? I’m sure it’s better understood now and one wouldn’t have to follow the historical approach to study the same concepts.

I don't know myself, but here are some possible answers:

https://mathoverflow.net/questions/114034/learning-path-for-...

Edit: Here is another overview that seems good:

https://www.math.ucdavis.edu/~osserman/math/pcm.pdf

Yeah, it's still incredibly difficult terrain, despite all the time that's passed. Wrestling with Hartsthorne is still a rite of passage for students in this area. I am (very) intrigued to know what would happen if someone made a serious effort to make the ideas more accessible. It remains an area where the depth of knowledge required is legitimately deep: would be cool to figure out which ideas are actually independent of other parts of the stack.

Take a look at the stacks project: https://stacks.math.columbia.edu

A cross between a wiki and a collaborative textbook.