An LL parse produces a leftmost derivation, which means that at each point in the derivation, the leftmost non-terminal must be replaced by one of its productions. The issue is to decide which production to use.
A grammar is LL(k) if the production to be used in the leftmost derivation can always be determined by examining only the terminals prior to the first non-terminal and the $k$ following terminals in the input. (These will be the next $k$ symbols in the input if we're doing a left-to-right parse; the terminals prior to the non-terminal will already have been input.)
In your grammar, neither $S$ nor $A$ present any problems at all. $S$ only has one production, so there is no need to examine any terminal at all to make a decision. $A$ has two productions, but their first terminal differs so it is trivial to predict which one to select: if the next input symbol is $a$, select the first alternative; if it is $b$, select the second one.
$B$ also has two productions, but both of them start with a $b$. So we need to look at least one symbol further in the input. Now, what is the second symbol in a derivation starting with each production for $B$? For $B \to b c$, the second symbol is clearly $c$. In $B \to b B c$, the second symbol must be the first symbol in some derivation of $B$. But, as we've just observed, every derivation of $B$ starts with $b$. So the second symbol in $B \to b B c$ must be a $b$.
With that, we have a complete decision procedure:
- If the non-terminal to expand is $S$:
- choose the production $S \to a a A$
- If the non-terminal to expand is $A$:
- If the next input is $a$:
- choose the production $A \to a A c$
- If the next input is $b$:
- choose the production $A \to b B c$
- If the non-terminal to expand is $B$:
- If the next two input symbols are $bb$
- choose the production $B \to b B c$
- If the next two input symbols are $bc$
- choose the production $B \to b c$
- If there is no non-terminal left
- If none of the above rules apply
The longest sequence of terminals we need to examine in any of those rules is 2. So the grammar is LL(2).