The article you linked does explain itself in more detail if you keep reading but to put it simply they are half the highest weights. The factor of a half is a physics convention.
The reason for the physics convention (I believe) is as follows. Consider representations of $\mathfrak{su}(2) \cong \mathfrak{so}(3)$. In maths we list the representations of this Lie algebra by their highest weight. Since the rank of these is 1, the weights live in a 1-dimensional space i.e. they are all scales of each other. In fact they are all positive integer scales of the "fundamental weight" so we can represent all irreducble representations by a positive integer. An important question is then which of these Lie algebra representations are also representations of $SO(3)$ and the answer is the ones with highest weight an even multiple of the fundamental weight. To make this distinction more stark, physicists halve the numbers so a representation extends to one for $SO(3)$ if the representative number is an integer. Then $\mathfrak{sl}(2,\mathbb{C})_\mathbb{C}$ is just two copies of $\mathfrak{sl}(2,\mathbb{C}) = \mathfrak{su}(2)_\mathbb{C}$ so its irreducible representations are just tensor products of an irreducible representation for each copy so can be represented by two integers (or two half integers in the physics system).
