Understanding spin both conceptually and mathematically...what are we measuring when we measure spin, and what justification do we have for calling it spin?
I think the confusion comes from the name. Suppose energy was named for what was applied originally to kenetic energy: maybe “speed” to pick something, that also has strong classical every-day meaning connotations.
Now energy comes in many forms, and explosive TNT or RDX can cause things to move and contains 'speed' in the chemical bonds. But the atoms are not moving around at high velocity but just sitting there in the solid. The bond, in particular, is a relationship and not an object per-se, so how can it be moving around to contain all that stored 'speed'?
Energy is a concerved quantity due to the symmetey of time. Any rules of nature that can be formulated using the action principle and respect the time symmetry (that is, the rules aren't different on Saturday) will conspire to keep this concerved quantity somewhere. Anything that changes the total kenetic energy must also change something else, and that something else can later be used to get back the same kenetic energy, and all the ways of holding onto some state that so interacts will agree in the end, like double-entry book-keeping. Knowing one obvious expression of energy, and the fact that it is concerved, you can tease out all the different forms of energy by the above principle.
Spin works the same way. The name causes confusion because it is named for one thing which later turned out to be accounted together with totally different things. So just call it AM here to try to avoid classical connotations.
AM is a concerved quantity due to the symmetry that space is the same in any direction. A quantity derived from masses moving around each other (including atoms on opposite sides of a baseball, so, it applies to rotating macroscopic objects) is an obvious form of AM, but not the only one. Anything that can change the above quantity must also capture the AM in another way, with consistent rules that ballance the books.
So, of note, electromagnetic fields can contain AM (and there is nothing there to move in a circular manner, and nothing changing over time at all! If energy were called speed, chemical bonds would give the same confusion), and elementary particles can contain AM.
Following the money, it makes sense that fields must represent an account in which AM can be deposited, because motors anf generators work, and can be so large that there is a delay due to light speed between the magnets and the load being torqed.
But why elementary particles? How do you exchange that with other forms at all, and why is it necessary for it to be there? The answer is the symmetry of spacetime. Because time and space are mixed and movement causes your time to spill into my space, you find that a combination of pure boosts (linear pushes) don't always result in a combined boost vector, like it does in classical space and (separate) time. In general, it may produce a boost and a rotation. Whether something is rotating or just moving straight can depend on your reference frame! This means that when two electrons (say) move past each other, there is some ambiguity over how much AM is present in the motion between them, and how much is spin. I don't think that last part is clear, but there are drawings and this proper explaination in a wonderful book The Force of Symmetry.
More authoratively, Richard Feynman remarked in his presentation of the 1986 Dirac Memorial Lectures, “At first [Dirac] thought that the spin, or the intrinsic angular momentum that the equation demanded...” Earlier attempts to figure it all out had spin as some feature that was put in because of the observations. But the new breakthrough formulas did not: the quanity representing spin popped out of the equasions; it was a necessary conclusion because (only!) of the symmetries going in. It's the bookkeeping again, and you could hope to understand the necessity by following a chain of rules that start with rotating motion being changed and following the money, where the AM must be deposited for the books to ballance in all possible uses of the rules. Combining relativity (spacetime symmetries) and quantum stuff makes intrinsic AM just show up if you work the math from first principles. Just like "why is 17 prime?" Thats the result you happen to produce starting with a few axioms and logic.
I hope that helps with the concept and justification side. Working the math is explained in other answers and you have a handle on that.