The Life of Swami Vivekananda by his Eastern and Western disciples - Vol 1
pag 46 of book (57 on pdf), final of 1st paragraph
Thus he went on amid sobs. The next moment he stood before me with
folded hands and began to address me, ‘Lord, I know you are that
ancient sage, Nara—the Incarnation of Narayana—born on earth to remove
the miseries of mankind,' and so on!
So, Sri Ramakrishna recognize him as Nara in his previous life (from Nara-Narayana twins), you can read about them on this question from Keshav Srinivasan on this forum
After that (Swamiji being Nara himself), you can keep tracking some lives
About why was he silent about his past lives? I don't know, it will be a conjecture. Some gurus say is not good to speak about one own spiritual progress, but I have not quotation, I'm pretty sure Rakhal (Swami Brahmananda) say something like this in a book titled "The eternal companion" but I can't find it now.
I believe his sayings was about some experience he attain as a boy, and his own mother tell him not to speak about that because he can "diluted" it and finally loose it, and Sri Ramakrishna confirming that. But not quotation for now.