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See Warren P. Johnson, The Pfaff/Cauchy derivative identities and Hurwitz type extensions, The Ramanujan Journal 13 (2007) pp. 167–201, link.springer.com/article/10.1007/s11139-006-0246-0, or my survey paper on Lagrange inversion, Journal of Combinatorial Theory, Series A 144 (2016) pp. 212–249, arxiv.org/abs/1609.05988, section 2.6.
This is a consequence of the continued fraction for the ordinary generating function for the numbers $I_k$, or (almost equivalently) that the numbers $I_k$ are moments for suitably normalized Hermite polynomials. See Krattenthaler's two papers on "Advanced Determinant Calculus": arxiv.org/abs/math/9902004 and arxiv.org/abs/math/0503507.
The identity follows by setting $x=1$ in the well-known formula $$\sum_{j=1}^\infty \frac{(xe^{-x})^j j^{j-1}}{j!} = x,$$ which is equivalent to the Taylor series for the Lambert W function that Robert Israel mentioned. A simple proof is by expanding the left side in powers of $x$ and using the fact that the $n$th difference of a polynomial of degree less than $n$ is 0. Another simple proof was given by Noam Elkies, math.harvard.edu/~elkies/Misc/abel.pdf.
I've recently written a survey paper on Lagrange inversion which can be found at arxiv.org/abs/1609.05988. You might also look at Sateesh Mane's paper at arxiv.org/abs/1607.04144. He gives a detailed account of the use of Lagrange inversion in solving algebraic equations.
What I had in mind is $$ \begin{aligned} \sum_{k=0}^n(-1)^k \frac{a}{k+a} \binom nk &= \sum_{k=0}^n (-1)^k a\binom nk\int_0^1 t^{k+a-1}\,dt\\ &=\int_0^1 a t^{a-1} \sum_{k=0}^n (-1)^k \binom nk t^k\,dt\\ &=a\int_0^1 t^{a-1}(1-t)^n\, dt\\ &=aB(a,n+1). \end{aligned} $$
We have $$\sum_{k=0}^n \frac{(-1)^k}{2k+1} \binom nk =\binom{n+1/2}{n}^{-1}.$$ This is equivalent to the case $a=1/2$ of the well-known identity $$ \sum_{k=0}^n(-1)^k \frac{a}{k+a} \binom nk =\binom{n+a}{n}^{-1} $$ which can be proved in many ways (e.g., induction, partial fractions, the beta integral, a special case of Vandermonde's theorem).
