There's a few points:
1. Julia is entirely dynamic, so there's no need to think about compile time versus run time, simplifying the mental model (but the performance is like that of compiled languages). It's as easy as Python or Matlab in that respect, but much faster.
2. There are just a few powerful language features (e.g. ubiquitous, fast multiple dispatch, supported with an expressive type system), rather than a lot of features that interact in complicated ways.
3. Good for general programming stuff: working with strings, calling external programs and other things that are generally pretty awkward in R and Matlab (one of the reasons why NumPy is gaining popularity).
In general, the motivation (expressed in a previous Julia blog post) is to have something that's easy to use and learn, but fast and powerful (you *can* go deep if you want to), and designed expressly for numerical work —which means, among other things, that it has to be able to store large arrays of numeric values in-line and call libraries like LAPACK on them.