In the solution they explain why they have to conclude that there should be at least one. Let me try again with a bit more explanation.

1. The liar stating something must mean that the phrase is not true. They cannot state anything that is not false.

2. "All X are Y" is the phrase.

Now, if we assume there is no X the phrases "All X are Y" and "Not all X are Y" are both true and false.

All X are Y - True. Yes, there is no X that is not Y.

All X are Y - False. Yes, there is no X that is Y.

Not all X are Y - True. Yes, there are no X, so none is not Y.

Not all X are Y - False. Yes, there are no X, so none is Y.

All these statements are (according to the article) vacuous if there is no X. A liar then cannot make them, as they are not false.

So from here you can deduce that either the phrase "All X are Y" stated by a liar indicates the existence of X or that I'm a liar :)