An LL grammar is a grammar that can be parsed by an LL parser. An LL parser is a parser which parses input from Left-to-right and constructs a Leftmost derivation of the sentence. (https://en.wikipedia.org/wiki/LL_grammar).
The explanation is still a bit confusing, so it's easier to think of it that by looking at only a portion of the input string, we can predict exactly which production rule to apply.
In the grammar you've given: $L(aa^*ba) \cup L(abbb^*)$
which has the production rules:
$$S \rightarrow aAba | abbB$$
$$A \rightarrow aA | \lambda$$
$$B \rightarrow Bb | \lambda$$
this grammar is the union of all strings which start with either an arbitrary amount of $a$'s or an $ab$.
Remember what I mentioned earlier about LL grammars parsing strings from left to right, and that we'd be able to predict which production rule to use by looking at a portion of the input string? This specific grammar satisfies these properties, and is therefore LL.
Let's look at an example:
Say we're given the strings $aaaaaba$ and $abbbbb$. Starting from the left, how many input characters must we read to see which production rule to use?
The answer is two.
All strings in the grammar $L(aa^*ba) \cup L(abbb^*)$ start with an $a$, and if the second character of the input string is also an $a$, then we must use the production
$$S \rightarrow aAba$$
and if the second character is a $b$ then we must use the production
$$S \rightarrow abbB$$
The only basic concept to keep in mind when dealing with LL grammars is whether or not you can accurately predict which production rule to use when only looking at a portion of the string.
I hope this helped, and if I either missed something or didn't explain sufficiently please feel free to correct me.