Merge sort consists of two main parts — diving part and merging part.
In the dividing part, the n-element array gets divided into two halves, and the halves into halves, ... until you have n one-element arrays.
As n grows, the number of steps to complete this process grows logarithmically. In this case, log base 2 of n (or just log n). Ex: it would take you 3 steps to divide an 8-element array and 4 steps to divide 16-element array.
Then comes the merge part in which each two halves get merged together. The number of steps to accomplish this task grows proportionally to n as n grows. So the runtime of this process is linear — O(n). Ex: if each half is 2-element long, it would take you typically 3 (i.e., n - 1 or roughly n steps, ignoring the -1) while it would take you roughly 8 steps to merge two 4-element arrays.
This process happens at every level (i.e., every step) of the dividing part.
Recall that the runtime of the latter is log n. This gives a total of n log n.