Method 1: characteristic functions
Referring to (say) the Wikipedia article on the multivariate normal distribution and using the 1D technique to compute sums in the article on sums of normal distributions, we find the log of its characteristic function is
$$i t \mu - t' \Sigma t.$$
The cf of a sum is the product of the cfs, so the logarithms add. This tells us the cf of the sum of two independent MVN distributions (indexed by 1 and 2) has a logarithm equal to
$$i t (\mu_1 + \mu_2) - t' (\Sigma_1 + \Sigma_2) t.$$
Because the cf uniquely determines the distribution we can immediately read off that the sum is MVN with mean $\mu_1 + \mu_2$ and variance $\Sigma_1 + \Sigma_2$.
Method 2: Linear combinations
View the pair of MVN distributions as being a single MVN with mean $(\mu_1, \mu_2)$ and covariance $\Sigma_1 \oplus \Sigma_2$. The sum is given by a linear transformation and therefore is MVN. (See p. 2 #4 here.) The covariance again works out to $\Sigma_1 + \Sigma_2$.