How do black holes break the laws of physics?
Black holes are extreme objects which can't be fully explained by the laws of physics as we know them.
A law of physics describes some physical property subject to some constraints. For example, Newton's laws of motion are only valid if objects are not travelling too fast or are not too massive. In fact Newton's laws are a good approximation to General Relativity for relatively slow moving and light objects.
Two great theories of physics are General Relativity and Quantum Mechanics. They both explain things very well. They are also both incomplete as the two theories can't be unified.
Black holes were first predicted by the Schwarzschild solution to General Relativity. Many people didn't believe they existed until evidence was found which could only be explained by a black hole. Some stars at the centre of our galaxy are travelling at speeds which can only be explained by them orbiting a body around the size of our solar system which weighs 4,000,000 times the mass of the Sun. Only a supermassive black hole fits this description.
Our current thinking about a black hole is that there has to be a singularity inside it. A singularity is a point of infinite density and infinite curvature of space time. The singularity simply can't be described by any physics as we know it.
The second problem with black holes is the information paradox. If something falls into a black hole and is destroyed. The information about its state is lost. This is not permitted by physics as we know it. Stephen Hawking is working on a theory that the information is somehow encoded in the event horizon which may solve this problem.
So, two ways by which black holes break the laws of physics as we know them are the singularity and the information paradox.