Q:

What does "keen" in "keen intellect" mean?

her keen intellect

What is the meaning of keen in the phrase?

A:

Keen is used in this sense as a stand-in for sharp.
See usage 1a at Merriam-Webster, for instance.
The usage "keen" for "sharp" is fairly common: "a keen edge", "a keen blade", etc.

A:

Although this does not represent the typical meaning of keen (eager is the more usual interpretation), the etymology of the word shows that the word has kept its ancient meaning in this case, meaning "knowledgeable" or "skillful".
To quote the Wiktionary page:

From Middle English kene (“bold,
  brave, sharp”), from Old English cēne
  (“bold, brave”), from Proto-Germanic
  *kōnjaz (“knowledgeable, skillful, able”), from Proto-Indo-European
  *g'enə-, *g'nō- (“to know”)

