Translation:He will continue on with the book.
My complaint is that the "lean ar" construction seems to only have appeared in this exercise. Checking in the Wingleacht foclóir, shows me that "leanfaidh é ar" is "He will continue/proceed with/persist in..." with "aghaidh" added to indicate "on/onward" or "forward". Wouldn't it be nice to have a past tense exercise like "Leanamar ar aghaidh go dtí an chéad sráidbhaile eile" to help work this into our brains?
I do not believe that "continue on" is a normal phrasal verb (at least in American English) — usually just plain "continue" would suffice. There is the phrase "carry on", however — might this not be a better translation of the intended meaning here? Of course "carry on" has a second, totally different sense too.