Finding strings that are matched by a given regular expression
Let's break down the regular expression $(c \cup ab \cup b)^*bb(c \cup ab)$. It has three parts $(c \cup ab \cup b)^*$, $bb$, and $(c \cup ab)$. The first part means "zero or more repetitions of $(c \cup ab \cup b)$", the second part matches two $b$, and the third part matches $c$ or $ab$.

As you already mentioned in your question bbc and bbab are strings that are matched. In both we choose zero repetition of the first part then the obligatory bb and then c, respectively ab, to match the third part. To make this easier to see I'll separate the parts of the string that are matched by the different parts of the expression with a - like so: -bb-c and -bb-ab.

We can generate more strings by choosing more than zero repetitions of the first part. For example we can choose one repetition of c to get c-bb-c. c-bb-ab would also be ok. We could also choose more repetitions of c or a c followed by ab for the first part. So cc-bb-c and cab-bb-c would also be acceptable strings.