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This is my context:

Sculptures must, for example, be stable, which requires an understanding of the properties of mass, weight distribution, and stress. Paintings must have rigid stretchers so that the canvas will be taut, and the paint must not deteriorate, crack, or discolor. These are problems that must be overcome by the artist because they tend to intrude upon his or her conception of the work. For example, in the early Italian Renaissance, bronze statues of horses with a raised foreleg usually had a cannonball under that hoof. This was done because the cannonball was needed to support the weight of the leg. In other words, the demands of the laws of physics, not the sculptor’s aesthetic intentions, placed the ball there. That this device was a necessary structural compromise is clear from the fact that the cannonball quickly disappeared when sculptors learned how to strengthen the internal structure of a statue with iron braces (iron being much stronger than bronze)

What does "that" in "That this device was..." refer to?

I find it weird to use "that" right before "this".

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    It doesn't refer to anything. "That" here is a complementizer -- it converts the sentence "this device was a necessary structural compromise" into a noun phrase, so that it can be the subject.
    – Greg Lee
    Dec 25, 2015 at 2:30
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    Try replacing "that..." with "the fact that...." It means the same thing and might help you make sense of the more concise version.
    – phoog
    Dec 25, 2015 at 5:28

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It's not anaphoric; it doesn't refer to anything. "That" is a subordinator introducing the subordinate subject clause "That this device was a necessary structural compromise". If you extrapose the subject clause, it becomes clear that "that" is a subordinator:

"It is clear from the fact that the cannonball quickly disappeared when sculptors learned how to strengthen the internal structure of a statue with iron braces (iron being much stronger than bronze) that this device was a necessary structural compromise".

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