。
It seems she likes talking about herself.
Literal
She [topic-は] her own self [about-のことを] talk [nominalizer-の] [subject-が] is-fond-of [seems-らしい].
~らしい ('apparently, it seems') marks an inference based on indirect evidence — observed behavior, hearsay, or general impression. Distinct from ~ようだ (more direct evidence) and ~みたい (casual). Here it lets the speaker characterize someone else without claiming direct insight: 'she comes across as someone who likes to talk about herself.' 自分のこと ('about oneself') is the standard frame for self-directed conversation topics.