This is a good example of why these sorts of pronoun choices are tricky.
It is also a good demonstration of why they are important, because in this case both constructions are grammatical and each refers to a different, but equally valid scenario.
Let's look at the first:
She’s more likely to ask him than I (am).
In order to focus on the choice at hand without the confusing presence of other pronouns, let's say:
Gia’s more likely to ask Jim than I.
This means it's more likely that Gia will ask Jim something than it is that I will ask Jim something. Both Gia and I can be thought of as subjects in this case (we're doing all the asking).
Then we have the second case:
She’s more likely to ask him than (ask) me.
Gia’s more likely to ask Jim than me.
This (as you know) means that, given the choice between Jim and myself, Gia is more likely to ask Jim something than she is to ask me something.
The key distinction between the two is that choosing “I” as the self-reference in the sentence groups the speaker with the people taking the action of asking, and choosing “me” groups the speaker with the people who are hypothetical targets of a question.
Implicit in the author’s note that the “distinction should be clarified with a revision” is the fact that this is something that many native speakers mess up all the time, choosing between subject/object pronouns in a casual way and relying on context to convey their meaning.
On a personal note, this is an aspect of English that my non-native-speaking friends and relatives have caught me doing wrong and corrected me on. Chances are, if you learn this well, you will outperform the average native speaker in selecting the appropriate pronoun.