How to get used to matricial/vectorial operations?
tl;dr: refresh your linear algebra and check for built-in functions.
In addition to brevity/elegance, vectorization often dramatically improves performance. Tools like MATLAB and numpy use high-level vectorized wrappers for low-level optimized implementations, which in the end are composed of your traditional loop operations. Where MATLAB and Python are interpreted (nuances exist, but are irrelevant here), the low-level subroutines that they wrap are obviously much faster. Also, tools that they wrap, like BLAS (Basic Linear Algebra Subprograms), are so heavily optimized that it'd be difficult for someone write a low-level subroutine that matches or exceeds the performance achieved with such libraries; many people over many decades worked to optimize them!
Considering problems from a math-oriented perspective helps with vectorization. For example, say you wanted to find the $L_p$ norm of a vector $\vec{x}$. A traditional CS program would look something like:
norm = 0
for i in vec_x:
    norm = norm + pow(i,p)
norm = pow(norm,1/p)

but a mathematician would simply write $||\vec{x}||_p$. First, you should check to see if a built-in function exists to perform your task, as a mathematician might. Then you should look to vectorize using math operators such as matrix multiplication, element-wise operations, etc., which these high-level tools implement. Going back to the $L_p$ example, you could write vectorized code (in MATLAB): (sum(x.^p))^(1/p), but it'd be even better to write norm(x,p). Knowing that such functions exist is a combination of familiarity with the tool and Googling.