Let the $5$-element subset be

$$a_1 < a_2 < a_3 < a_4 < a_5$$

with each $a_{i+1} \ge a_i + 2$ (no two elements are consecutive).

Define

$$b_1 = a_1,\quad b_2 = a_2 - 1,\quad b_3 = a_3 - 2,\quad b_4 = a_4 - 3,\quad b_5 = a_5 - 4.$$

Because each $a_{i+1} \ge a_i + 2$, the sequence $b_1 < b_2 < b_3 < b_4 < b_5$ consists of distinct integers from $\{1,2,\dots,11\}$.

Thus, there is a bijection between:

$$\{\text{$5$-element subsets of $\{1,\dots,15\}$ with no consecutive integers}\}$$ 

and

$$\{\text{$5$-element subsets of $\{1,\dots,11\}$}$$