What does {a,b}* for DFA's mean?
$\{a,b\}$ is set notation. The alphabet of a language is the set of all possible symbol used in the language. You can substitute the typical $\Sigma$ for the full set every time.

$\Sigma^*$ (where $^*$ is the kleene star) is the set of all possible strings you can make with that alphabet. That includes strings that happen to not use a certain character.

Therefor $\{w \in \{a,b\}^* : w \text{ contains } bbb\}$ is equivalent to $\{w \in \Sigma^* : \Sigma =  \{a,b\} \text{ and } w \text{ contains } bbb\}$. But it skipped defining $\Sigma$ for terseness reasons.