Why doesn't gravity fit into quantum theory?

I'm going to give an answer that is not complete, but maybe gives a flavor for why gravity is different than the other forces. Note to experts that I know that I am oversimplifying in this answer but the OP explicitly asked for an answer that aimed at simplicity, I will try to point out some of the simplifications I made at the end.

Electromagnetism is the classic example of a well behaved force where we know how to treat it quantum mechanically. Gravity, on the other hand, is famously ill behaved.

So a different way you might ask your question is, why is gravity so much different than electromagetism?

One reason (and this is not the full story by any means) is that electromagnetism, unlike gravity, obeys the superposition principle. If I have two point charges, the compute the electric field of the two charges I can simply add the electric fields that each charge produces individually to find the total electric field of the combination of the two point charges.

Gravity is not like this. If I have two point masses (black holes), and compute the gravitational field of them individually, I can't simply combine those two fields to produce the gravitational field of the two masses together. The basic reason is that because $E=mc^2$, not just mass but also energy gravitates. The gravitational field itself carries energy. So the mass of one black hole creates a gravitational field, but that field has energy, which itself sources more gravitational field, and so on... When you have two point masses, there is interaction energy as the masses attract each other, and that interaction energy itself changes the gravitational field. It becomes very complicated very quickly! The basic fact that all forms of energy and momentum--including the energy carried by the gravitational field itself--gravitates, makes gravity intrinsically more complicated than electromagnetism.

To get ever so slightly more technical, because of the superposition principle (and some other nice properties like Poincaire invariance) you can think of the electromagnetic field as being composed of harmonic oscillators (one for each fourier mode of the field). We know how to quantize harmonic oscillators! For gravity this treatment doesn't work, because of the failure of the superposition principle. Now, when the gravitational fields are small (meaning, when the curvature is small, or really when you are working on distances small compared to the characteristic curvature scale), you can try to approximate the gravitational field as being made of harmonic oscillators. And that is a consistent procedure when the gravitational fields are weak--this is called the effective field theory of gravity--although it is doomed to break down when the gravitational fields become too strong (meaning in practice that the curvature is large). That's really where many of the hard problems with quantum gravity kick in.

Now like I said I am definitely oversimplifying, for example experts may complain that (1) I have used the phrase 'gravitational field', (2) that I've talked about the energy of the gravitational field, and (3) what about Yang-Mills which is an example of a theory which doesn't obey the superposition principle but which is perturbatively renormalizable, and possibly even (4) do we really know the effective field theory of gravity works for example near black hole horizons (what about the firewall paradox and complementarity)? All these objections (and I'm sure others) are perfectly valid (except maybe for (1) but I don't want to start a fight :)). All I'll say is that this answer is just meant to give you a rough intuitive flavor of some of the difficulties going on--there is a lot, lot more to the story.

I'll just end by saying that a more correct answer is that gravity is mediated by a spin-2 field whereas electromagnetism is mediated by a spin-1 field. The "spin-2"-ness really is what ends up making gravity so much different, from a particle physics perspective.