In terms of hypergeometric series, the sum is $_3F_2(-n, 1+n, 1/2;1,3/2;1)$ and the identity is a special case of [Saalschütz's theorem][1] (also called the Pfaff-Saalschütz theorem), one of the standard hypergeometric series identities. 

A more general identity, also a special case of Saalschütz's theorem, is 
$$\sum_{k=0}^n (-1)^k\frac{a}{a+k}\binom{n+k+b}{n-k}\binom{2k+b}{k}
  = \binom{n+b-a}{n}\biggm/\binom{n+a}{n}.$$
The O.P.'s identity is the case $a=1/2, b=0$.


  [1]: https://en.wikipedia.org/wiki/Generalized_hypergeometric_function#Saalsch%C3%BCtz's_theorem