。
She likes being all by herself.
Literal
She [topic-は] all-alone [as-で] being [nominalizer-の] [subject-が] is-fond.
一人っきり intensifies plain 一人 with the gemination っ + 切り — 'completely on one's own, alone with no exception.' The っき form is colloquial; written variants include 一人っ切り and 独りっきり (the 独 spelling carries a slightly more emotive 'solitary' tone). 居る ('to be, to exist for animate beings') marks the state of presence; ~のが好き makes the experience of solitary being itself the object of fondness.