What exactly is the LL(k) grammar condition?
It's actually a definition of a strong LL(k) grammar.

First of all, see Raphael's comment.

In short, the condition means that while parsing $A$ you can choose the next production rule in a deterministic manner using only the next $k$ lookahead symbols.

An important thing is you don't need any context for that decision, i.e. you don't need to remember the previous steps in the left-most derivation that led you to the non-terminal $A$ to parse $A$ itself.

By the way, every LL(1) grammar is strong, but there are non-strong LL(k) grammars, $k &gt; 1$.

If you need a more detailed explanation, then I think it's better to use a book for that (it's too long for an answer here). I'd suggest "Parsing Techniques - A Practical Guide" by Dick Grune et al.