Chinese restaurant process vs Dirichlet Process
A sample from a Dirichlet Process, or DP, is a distribution over a sample space $S$. Here the DP is defined based on a base distribution $H$ over $S$. For instance, in the Wikipedia example from your page, the sample space is all real numbers $\mathbb{R}$ and the base distribution is the standard Normal. 

The Chinese restaurant process, or CRP, defines a partition over integers $1,2,...,n$ at each time $n$, and $n$ can go to $\infty$. In this metaphor each block in the partition is called a table. Notice that the CRP itself has nothing to do with the original sample space $S$ or the base distribution $H$.

To associate draws from the base measure H with every table, it means for each table $i$, you independently draw one sample $s_i$ from $S$ according to the base distribution $H$. You repeat the sample $s_i$ for $b_i$ times, where $b_i$ is the number of "customers" seated in the $i$'th table in that Chinese restaurant. The distribution of all those repeated samples for all tables, is your sample distribution from the DP.