Efficient algorithm for finding maximum subset of intersecting rectangles
You need a sweep line and an interval tree (modified so that it reports a number instead of intervals). I assume this is a course book problem and you are familiar with its background.

Move the sweep line bottom to top, stop at each horizontal segment. If the segment is the bottom side of a rectangle, insert it as an interval into the interval tree and remove it when the sweep line hits the top side of that rectangle. At each step you can query the number of rectangles containing a point on the sweep line. There are $O(n)$ stops for sweep line, each one adds $O(\log n)$ for interval operations, so it is $O(n\log n)$ in total. The only remaining issue is to keep track of the maximum covered point in $O(1)$ at each step. You should be able to complete this solution.