This identity can be written in the following form
$$\sum_{i=0}^k (-1)^{k-i+1} B_i\cdot\binom{k+1}{i}=(-1)^{k+1}(k+1).$$
The latter is equivalent to $B_{k+1}(0)-B_{k+1}(-1)=(k+1)(-1)^k$ which is one of the basic properties of Bernoulli polynomials.