$H(B_i)$ is the probability of the set $B_i$ under $H$, ie, if $Z\sim H$, $H(B_i)\equiv P(Z\in B_i)$.
$(X(B_1),\dots, X(B_n)$ is a random vector. It has a Dirichlet distribution, so it is a set of proportions adding up to 1, which makes sense since the $B_i$ are a partition adding up to the whole space.
The means of the $n$ proportions are proportional to the $H(B_i)$ (but add up to 1), and as $\alpha$ increases, the variances of the proportions get smaller. For large $\alpha$, $X(B_i)$ will be close to $H(B_i)$, but for small $\alpha$, $X(B_i)$ will be vary a lot.
One way to think of the Dirichlet distribution is that the proportions are as close to independent as they could be given that they have to add up to 1 (you can simulate a Dirichlet distribution by simulating independent Gamma distributions and dividing them by their sum)