The usual definition of an algebra being generated by a subspace is as follows:
Let $A$ be an algebra, $X \subset A$ a subspace, $\mathrm{Alg}(X)$ the free algebra generated by $X$. Then $A$ is generated by $X$ iff the morphism $\mathrm{Alg}(X) \to A$ induced by the inclusion $X \hookrightarrow A$ is a surjection.
The interpretation here is clear: every element of $A$ can be written as a finite linear combination of "words" of elements of $X$.
Now the dual statement is (see eg. Operads in algebra, topology and physics by Markl et al.):
Let $C$ be a coalgebra, $Y$ a quotient of $C$, $\mathrm{Alg^c}(Y)$ the cofree coalgebra cogenerated by $Y$. Then $C$ is cogenerated by $Y$ if the morphism $C \to \mathrm{Alg^c}(Y)$, coinduced by $C \twoheadrightarrow Y$, is injective.
I don't know how to interpret that. If we restrict ourselves to conilpotent coalgebras, we can take the cofree conilpotent coalgebra generated by $Y$ to be $T^c(Y)$ the usual tensor coalgebra; $T^c(Y) \to Y$ is the projection on the $Y$ factor, and we have an explicit description of the morphism $\tilde p : C \to T^c(Y)$, with the projection of the $Y^{\otimes n}$ factor is $\sum p(x_{(1)}) \otimes \dots \otimes p(x_{(n)})$. Somehow if $Y$ cogenerates $C$, then this is injective. Which would roughly mean that for any $x \in C$, if you iterate the coproduct enough time, the factors are nonzero through the projection on $Y$. What does that mean? Is there a way to interpret that?
I can't comment under DBr's answer for being a noob. But at least in the conilpotent case (maybe also pointed coalgebra case), he has answered your question: every element of $C$ is a sum of words in $Y$ freely joined by $\otimes$.
Just think in terms of combinatorics: comultiplication codes the deconcatenation of words ($abcd \to a | bcd + ab | cd + abc | d$ using reduced $\overline{\Delta}$) which is dual to concatenation of words coded by multiplication ($a * bcd \to abcd$ etc). You can think $C$ is formed by words in $Y$ but you are looking at deconcatenation process in the algebraic structure.
Edit: Not every words will be in C so I deleted "freely generated".