'keen' has the sense of 'very much/deeply intellectually interested and 'revving' to pursuing some interest or activity'
'eager' adds to this a sense of emotional excitement and enthusiasm, with a touch of 'can't get enough' of it.
If these words were being used appropriately, then in your sentence
She sounded very keen to meet you - ( because you both share an interest, or know a lot more about some subject or have an especial skill in an activity.)
She sounded very eager to meet you.- (because from what she has heard about you, you may be the 'man of her dreams' perhaps?)
Because 'eager' is the more emotionally based of the two, the sense of eagerness is of a shorter duration, just as the intensity of an emotional feeling is of a shorter duration than an intellectual interest. Hence, I can be a 'keen gardener', indicating a hobby pursued over many years; but not 'eager gardener'. But I can be 'eager' to get started on something which has fired my intellectual interest and has me emotionally excited. Once I start, the emotion will probably soon calm down, and I am left feeling 'keen' to keep going and do more.
Again, therefore, in terms of your sentence: I may be eager to meet someone, but if I find them dreary, I am probably not keen to pursue the relationship!