What "soft-skills" does one need to succeed as a mathematician?

A few that come to mind, and are probably valid in other fields also --

• Learn how to give good talks.

• Learn how to describe your research informally. Say you meet someone at a conference, and they ask you "What do you work on?" Can you convince them that what you work on is interesting?

• Learn how to come up with interesting questions that you don't know how to answer.

• Learn how to read a paper. Can you figure out "the big idea" without getting bogged down in the technicalities?

• Learn how to meaningfully participate in a math discussion when you're less than 100% sure what's going on.

• Develop a gut feeling for when a subject will come in useful. For example, "This lemma feels like I could use algebraic geometry to prove it", without knowing initially what that form would take.

• Get a sense for what other researchers in your field consider interesting. What is worth writing a paper about, and what is worth giving a talk about?

There are many, many more -- and none of the above are easy.

Understanding the research process and how to overcome hurdles and frustrations. Some of these points have already been addressed in other answers, but I think an explicit list of things around this is good. This means:

• Knowing how to find interesting research questions.

• Knowing how to get started on the research process.

• Understanding what it means to put time and effort into solving a research problem.

• Knowing how to find collaborators that would be interested in working with you, and then working with them in a productive manner.

... and many other things