How to interpret the breeders equation?
The breeder's equation as you wrote it:

$$R = h^2S$$

The heritability that is the ratio of additive genetic variance over the total phenotypic variance is called the heritability in the narrow sense and is noted $h_N^2 = \frac{V_a}{V_P}$, where $V_A$ and $V_P$ are the additive genetic and phenotypic variance respectively. In contrast the heritability that consider the total genetic variance is called the heritability in the broad sense and is noted $h_B^2 = \frac{V_G}{V_P} = \frac{V_A + V_D}{V_P}$, where $V_G$ is the total genetic variance and is equal the sum of the additive genetic variance $V_A$ and the genetic dominance variance $V_D$. We can rewrite the breeder's equation using the standard notation for the heritability in the narrow sense:

$$R = h_N^2S$$

The so-called response to selection $R$ is the mean deviation of the trait in the next generation. In other words, $R$ is the difference between the mean trait of the parents and the mean trait of the offspring, So to answer your question, yes, the breeder's equation gives the value of the mean trait in the following generation. It does not give any information about the frequency and number of alleles though. 

$S$ is the the mean trait of the individuals which reproduce weighted by their reproductive success minus the mean trait of the parents. In other words, it is 

$$S=\frac{1}{\bar w}\sum_{i=1}^{\text{nb inds}}m_iw_i - \bar m$$

, where $m_i$ is the trait measurement and $w_i$ is the fitness of this individual, $\bar m$ is the mean trait of the parents and $\bar w$ is the mean fitness.

The mean trait of the parent generation does not appear in the breeder's equation but only their mean trait weighted by their reproductive success. In the standard case of truncated selection (common in artificial selection programs) all $w_i$ are equal and $S$ becomes the mean trait of the individuals that we allow to reproduce in the parent population. And it is exactly the same thing for the multivariate case except that the variance-covariance matrix should be taken into account.