How do you make this kind of light through plastic material?

Mix diffuse/translucency, then with rough refraction using geometry/incoming normal. Then fresnel mix with glossy. At the end of it all, you may want to control how shadows are calculated, by mixing in transparency using light path/isShadowRay.

I halfway understand half of that. In your solution would light be emitted from the material or would there be a light object within the model? If there has to be a light within the model, does the model need to have thickness?

I tried it with a point lamp inside the model and no thickness.
First a simple setup using the principled BSDF:

Here is an attempt to do it the way CarlG described, though I’m unsure, whether I set it up right. I’m not really experienced constructing materials like this, but it definitely gives nicer results!

Looks about right to me. For a glossy roughness that high, you may want to consider rough fresnel.
It’s basically a replacement for the fresnel node so that the maximum fresnel output is lowered as roughness increases. Principled does this automatically, but principled doesn’t have translucency, and although it can be ignored, glossy should be mixed in at the end. Simply mixing translucency with principled will cause the glossy component to darken. Refraction is mostly there to provide a view dependent highlight if desired. For translucent plastic, I’d increase it. For paper, I’d lower it - possibly even turn if off. I setup rough fresnel like this:

The Diffuse is the rest of your setup, and the dummy is needed if you make a node group out of it, due to a bug in Blender which can produce garbage if input socket is not used.

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