I'm reading the paper Cardinal invariants above the continuum by J. Cummings and S. Shelah. There they have de following lemma:
I have a doubt about with the yellow part: what they really means with that? Because it is well know that every poset that adds a dominating real (e.g Hechler forcing) destroys every unbounded set of $(\omega^\omega,\leq^*)$ in the ground model.
I apreciate any aclaration about this point.
"Unbounded set" here means "unbounded subset of $\mathbb{P}$" - see Definition $1$. This definition only makes use of quantification over elements of $\mathbb{P}$, and so is absolute between $V$ and $W$ for any extension $W$.
(Note that it's crucial that we're working here with a poset and subset of that poset as opposed to a definition of a poset and subset. Actual sets don't change between extensions, but definitions can.)