Proof that quicksort's running time is ∼1.39 n log n
it's claimed that the running time of QuickSort is $∼1.39\,n\log_2n$


I don't have the book handy, but it it most certainly does not claim this. That figure is the result of a close analysis of a very specific cost measure (which "time" is not); expected number of comparisons, if I remember correctly.

The proof is given in the book, if I recall correctly.


  quicksort's average case is about $2n\log_e n$.


This is basically the same figure since $1.39 \approx \frac{2}{\log_2 e}$, the difference probably being a rounding artifact.

From your comment, quoting Sedgewick/Wayne:


  In summary, you can be sure that the running time of algorithm 2.5 (quicksort) will be within a constant factor of $1.39n \lg n$ whenever it it used to sort n items.


This only states that the (expected) running time is in $\Theta(n \lg n)$; this follows from the (expected) number of comparisons being what it is, plus that comparison is a dominant operation in Quicksort. 

For what it's worth, the formulation "you can be sure" without context is misleading since it seems to ignore the worst-case, and the asymptotic nature of the result.