This note says in the last sentence, "The discontinuities of multivalued functions in the complex plane are commonly handled through the adoption of branch cuts, but use of Riemann surfaces is another possibility."
I have never heard of this before. Can someone explain what Riemann surfaces have to do with multivalued functions? For example, can one explain the meaning and holomorphicity of a function like $f(z) = \sqrt[m]{z}$ in terms of Riemann surfaces instead of branch cuts?
Riemann surfaces become handy for multivalued functions because they help you to "transform" a multivalued function in a univalued function. Usually, the most common example is the m-th root of z. The idea is that for the m-th root of z is that there are m points whose m-th power is z. If you take these m points, they form an m-gon (if this is not so clear, do an example with the unity), that satisfies that each vertex has as its m-th power z. This implies that $z^m$ is not injective and this is the reason why $z^{1/m}$ is multivalued.
Now the set defined by the points between (in terms of angle and module) the lines that go form the center of the m-gon to two consecutive vertices till infinity, will satisfy that if you evaluate it in the m power, you'll get again the whole plane, so each "slice" of the m-gon forms a copy of the plane, and this is when Riemann surfaces become handy, because you construct them with these copies of the plane, you put one over the other one (kind of forming a parking lot of m floors).
I know this explanation is a really intuitive one, and maybe a bit oversimplified, but if you are more interested you can find information in the book "Introductory Complex Analysis" by Richard A. Silverman.