To the manner born comes from Hamlet and means something like accustomed from birth or born with a talent, proclivity or ability. So, our sentence means that although the CEO’s emotional intelligence was lacking, she worked on her people skills and improved.
The phrase is often thought to be to the manor born and is used similarly to another one: born with a silver spoon in his mouth — meaning born into a life of wealth and privilege.
However, to the manner born refers to innate ability (or proclivity or acclimatization) of any sort. As Random House’s Word Maven explains, the original reference is to the ability to drink copious amounts of liquor:
The first thing to note is that this is not some random phrase which we have to puzzle out, but a Literary Allusion, from the WotD’s favorite alludee, Shakespeare. In Hamlet, when the prince is observing the drunkenness common at Elsinore, he complains, “But to my mind–though I am native here,/And to the manner born–it is a custom/More honor’d in the the breach than the observance” (Hamlet I.iv.14ff).
What Hamlet meant here is that he is destined or accustomed from birth to the practice of heavy drinking. The phrase is often used in a sense like this (‘accustomed to (a specified practice) since birth’, that is), or often somewhat more broadly meaning ‘admirably well suited for (something specified)’.
So, in fact, you could say a champion yo-yoer, a virtuoso hog-caller or a talented pickpocket from a long line of talented pickpockets was to the manner born. And then, when you got some funny looks, you could explain…
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