The strange properties of half-integer spin are one of those mysterious facts that make physics interesting. It's part of the geometry of a mathematical object called a 'spinor'. The name comes from making an analogy with a 'vector', but related to spin.
Spinors are fairly advanced mathematics, but there is an intuitive way to think about them that works a lot of the time. A spinor can be thought of as an ordered pair of oriented reflecting planes passing through a point. 'Ordered' means that we distinguish the first and second plane. 'Oriented' means they have a front and a back face. If you do two reflections one after the other, you get a rotation, so any spinor has a rotation associated with it.
Now if the two planes are identical and facing the same way, the first and second reflection reverse one another, and you get the identity. As you twist one of the planes with respect to the other, the corresponding rotation is through twice the angle between the intersecting planes, about an axis along the line where the two planes meet. When the planes get to 45 degrees apart, they produce a 90 degree rotation. When they get to 90 degrees apart they produce a 180 degree rotation. When they are 135 degrees apart the result is a 270 degree rotation. And when the planes are 180 degrees apart, so they are lined up back-to-back, you get a 360 degree rotation. Note that because the planes have a well-defined front and back, two planes back-to-back are the opposite of two planes facing the same way.
The distinction is invisible if all you can see is the rotation that results, but the physics of electrons can tell the difference between planes back-to-back or facing the same way.
If you keep on rotating one plane with respect to the other, when the first plane has turned 360 degrees and is back to where it started, the rotation produced by the pair of reflections has turned a full 720 degrees. This is what is meant by saying you have to rotate a spinor 720 degrees to get back to where you started. It's the rotation represented by the pair of planes that is being spun.
I should say, the above picture is not quite right. The spinor is really representing some kind of oriented angle in space. It's the intersection axis and the angle between the planes that matters, not the particular planes themselves. So if you rotate both of the planes together about their common axis (rather than just rotating one with respect to the other) you get the same spinor. Similarly, any pair of perfectly aligned planes is the identity and always the same, and any pair of perfectly opposed planes is its negative. I realise that's probably somewhat unclear - it's hard to give an intuitive geometric picture of a very unfamiliar concept.
Now I'm going to get more technical, for anyone who wants to look into it more deeply. There is an extension to the geometry of vectors called 'geometric algebra', which was originally discovered by the English mathematician William Kingdon Clifford back in the days when they were inventing vector geometry. Clifford's geometric algebra was actually invented before vectors were fully developed and accepted, but vectors took over and Clifford's ideas were abandoned and forgotten in physics for a long time.
Geometric algebra has several different sorts of object in it, and unifies them all into a single structure. Scalars, which have no direction or extent; vectors, which represent a one-dimensional extent, a line, a directed length, or a reflection; bivectors, which represent a two-dimensional extent, a plane, a directed angle, or a rotation; and trivectors which represent a three-dimensional extent, or a volume. Scalars, vectors, bivectors, and trivectors can all be added and multiplied together freely, so you can add a vector to a scalar and then multiply the result by a bivector and so on.
The 'spinors' in any space are 'the even sub-algebra' of the geometric algebra: the elements made up entirely of even-dimensional components. So in 3D space, as described above, this consists of just the scalars (0-dimensional) and bivectors (2-dimensional) and linear combinations of those. And in 3D this happens to be the space of products produced by multiplying two vectors together, each vector representing a reflection. When you multiply vectors in geometric algebra, you get the sum of a scalar (the dot product) and a bivector (closely related to the cross product) that describe how the vectors are oriented with respect to each other.
In 3D, the spinors (scalars + bivectors) also happen to be the same as the 'quaternions', which you may also have heard of. In 4D and higher, things get more complicated.
Geometric algebra gives a much more intuitive, 'geometric' picture of spinors, which otherwise are just mysterious abstract entities that you have to take on trust without ever really understanding them, which is what most physicists do.