Merry New Year!

Take log for both numbers.

$log2015 = 3.3042750504771$

$log 2016 = 3.3044905277734$

$log(2015)^{2016}\ = 2016log(2015) = 6661.418501$

$log(2016)^{2015}\ = 2015log(2016) = 6658.5484134634$

Characteristic of $log(2015)^{2016}\ = 6661$

Characteristic of $log(2016)^{2015}\ = 6658$

Now, characteristic of a number = No. of digits + 1

As $log(2015)^{2016}$ has larger characteristic, So it has more no. of digits, so it is bigger.

$2015 < 2016 \\ 2015^{2015} < 2016^{2015} \\ 2015^{2015} \times 2015 > 2016^{2015} \\ 2015^{2016} > 2016^{2015}$