The answer depends on whether you are assuming the symmetric or asymmetric dirichlet distribution (or, more technically, whether the base measure is uniform). Unless something else is specified, most implementations of LDA assume the distribution is symmetric.
For the symmetric distribution, a high alpha-value means that each document is likely to contain a mixture of most of the topics, and not any single topic specifically. A low alpha value puts less such constraints on documents and means that it is more likely that a document may contain mixture of just a few, or even only one, of the topics. Likewise, a high beta-value means that each topic is likely to contain a mixture of most of the words, and not any word specifically, while a low value means that a topic may contain a mixture of just a few of the words.
If, on the other hand, the distribution is asymmetric, a high alpha-value means that a specific topic distribution (depending on the base measure) is more likely for each document. Similarly, high beta-values means each topic is more likely to contain a specific word mix defined by the base measure.
In practice, a high alpha-value will lead to documents being more similar in terms of what topics they contain. A high beta-value will similarly lead to topics being more similar in terms of what words they contain.
So, yes, the alpha-parameters specify prior beliefs about topic sparsity/uniformity in the documents. I'm not entirely sure what you mean by "mutual exclusiveness of topics in terms of words" though.
More generally, these are concentration parameters for the dirichlet distribution used in the LDA model. To gain some intuitive understanding of how this works, this presentation contains some nice illustrations, as well as a good explanation of LDA in general.
An additional comment I'll put here, since I can't comment on your original question: From what I've seen, the alpha- and beta-parameters can somewhat confusingly refer to several different parameterizations. The underlying dirichlet distribution is usually parameterized with the vector $(\alpha_1, \alpha_2, ... ,\alpha_K)$ , but this can be decomposed into the base measure $u = (u_1, u_2, ..., u_K)$ and the concentration parameter $\alpha$, such that $\alpha * \textbf{u} = (\alpha_1, \alpha_2, ... ,\alpha_K)$ . In the case where the alpha parameter is a scalar, it is usually meant the concentration parameter $\alpha$, but it can also mean the values of $(\alpha_1, \alpha_2, ... ,\alpha_K)$, since these will be equal under the symmetrical dirichlet distribution. If it's a vector, it usually refers to $(\alpha_1, \alpha_2, ... ,\alpha_K)$. I'm not sure which parametrization is most common, but in my reply I assume you meant the alpha- and beta-values as the concentration parameters.
