This is the 11429th most frequent Polish word.
intymnymi
"Intymnymi" means "intimate" (instrumental plural form).
In this sentence, 'intymnymi' is used as an adjective agreed with 'tematami' in the instrumental case, describing the nature of the topics as intimate.
Miałem okazję rozmawiać z nimi na bardzo intymnymi tematami.
I had the opportunity to talk with them about very intimate topics.
Here, 'intymnymi' modifies 'sekretami' also in the instrumental case, emphasizing that the secrets being mentioned have a private and personal nature.
Nie dzielimy się intymnymi sekretami z obcymi.
We do not share intimate secrets with strangers.
The adjective 'intymnymi' is paired with 'fragmentami,' again in the instrumental case, again to describe the segments of life as private and personal.