When your two eyes disagree sharply Now and then, the image from each eye is so different from the other that it seems the brain has trouble combining them into a coherent image. You can create this situation by putting a hand or sheet or paper in front of one eye, then moving it about an inch away. If you open one eye at a time, you see the obstruction with one and what it obstructs with the other. If you keep both eyes open, you see a composite where the obstruction appears translucent or transparent in parts.

I've been curious about this phenomenon. Is there a name for it? Is it clear why the brain handles it this way, producing a seemingly impossible object, and not some other way?
Binocular rivalry refers to a situation where you present different images to each eye (often in the context of an experiment, although you also point out some quick 'at home' ways to achieve this), and your perception switches between what each eye observes. 

This phenomenon is related to other forms of bistable/multistable perception, one of the most famous examples of which is the Necker Cube but there are many many examples. 

It's hard to answer your question about why this is the case, but I think it's helpful to think about what the purpose of the brain is: making a model of the world for you to act in. It doesn't really make sense for the world to be in two different states simultaneously. Your brain is constantly making perceptual decisions that translate observations into a model.

To make just a toy example for why you may want to select a single model rather than trying to incorporate both forms of information simultaneously, imagine you are some animal trying to catch a flying insect. One eye suggests the insect is in one part of space; the other eye suggests a slightly different part of space. You can reach to only one of the two spaces. You are more likely to be successful if you choose one of the targets (at least you have a 50/50 chance), whereas if you used some other strategy like an average of the two positions, you would miss every time.

There is also a lot of literature focusing on the role of attention in binocular rivalry as well as other aspects. It's probably beyond the scope of SE to provide a full review, but hopefully this is enough information for you to do some of your own further research on.