How do I interpret this formal languages definition?
It's the same $w$. You can think of this language in a ``procedural'' way: it is  formed by taking every word $w\in \{a,b\}^+$, and for each such word, adding to the language the word $wcw$.

Typically, sets (and languages in particular) are described as $\{x: \text{condition on } x\}$, which is pretty clear. However, it is sometimes more convenient to ``push'' the condition into the description of an element. 

That is, you could describe the same language as:
$$\{x\in \Sigma^*:\ \exists w\in \{a,b\}^+\wedge x=wcw\}$$
Or even worse:
$$\{\sigma_1\cdots \sigma_n: \exists k\ge 1,\ n=2k+1\wedge \sigma_1\cdots\sigma_k=\sigma_{k+2}\cdots\sigma_{2k+1}\wedge \sigma_{k+1}=c\wedge \forall 1\le i\le k,\ \sigma_i\in \{a,b\}\}$$

But obviously this is much more cumbersome than the original description.