Reason says the bubbles should go on the inside:

There are gaps between the bubbles. If they're on the outside, something can impact the item in a gap and bypass the padding. If the bubbles are on the inside, they become supports for the flat side (which, IIRC, is usually heavier plastic), so if there's an impact on a gap, there's still padding because the plastic is being held away from the item and the force is transferred to the surrounding bubbles.

That's how I was thinking about it before this article at least.

Imagine a small enough object, like a tip of a screwdriver, or a table's corner, touching an object wrapped bubbles out.

It simply makes contact with the object - here covered with only a very thin layer of plastic - between the bubbles, and given enough force, damages the object.

With bubbles inwards, it has to pierce or stretch the now airgapped layer of plastic first.

Even with multiple layers, bubbles inside give +1 airgapped layer.