What makes anything transparent is that light is able to pass through it without being scattered or blocked. Some materials, like metals are structured in such a way that they completely block the light, so you can’t see through it. Other materials, like many glasses, are structured so that the light passes through without being blocked or too distorted.
To a degree, thickness does affect whether you can see through an object. Even things like glasses have an upper limit to how thick they can be. Also, if you get thin enough, some opaque materials will become see-through, but those are at very small scales.
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