Why do objects with the same charge repel each other?

This is a great question, and there's at least two ways of answering it: an explanation of what we know about the physics underlying electromagnetic interactions, and a philosophical answer. 

As far as the philosophical answer, i.e. "why does anything do X," the answer is that this is the way our universe works. It's entirely possible that, in other "universes" at other times or under other conditions in our own universe, this would change. Without...

This is a great question, and there's at least two ways of answering it: an explanation of what we know about the physics underlying electromagnetic interactions, and a philosophical answer. 

As far as the philosophical answer, i.e. "why does anything do X," the answer is that this is the way our universe works. It's entirely possible that, in other "universes" at other times or under other conditions in our own universe, this would change. Without a deeper understanding of why the physical rules of our universe have the values they have, we can only speculate why anything is the way it is. It's frustrating but also interesting, at least to me, because you end up asking more questions and discovering more information you didn't know. 

The more technical answer has to do with quantum mechanics. Interactions between objects are typically mediated by particles called force carriers. For the electromagnetic force (i.e. charges), the force carrier is a photon. Photons also have a property called spin, which is one aspect of the way we describe the photon's existence as a "blip" in the electromagnetic field. Because the photon has a spin value of 1, like charges repel; for force carriers with spins of 0 and 2, like charges actually attract.

Why does spin 1 mean that like charges will repel? Getting into this level of detail would require a tutorial on relativity. To simplify, you can think of a charge as being a distortion in a field, either up or down (it's not literally up or down, but we can still make a point). The field wants to be flat. When opposite charges, i.e. opposite distortions, come in contact, they "flatten out," whereas similar distortions would add even more stress to the system. For systems where the spin of the force carrier is 0 or 2, you can think of it as NOT wanting to be neutral—it wants to stack up as many of those charges as it can. This is why gravity involves "like charges" (mass) attracting each other; because gravity can be thought of as having a spin 2 force carrier.