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She really has a firm grip on the public's hearts, doesn't she.
Literal
She [topic-は] public [genitive-の] hearts [object-を] firmly is-grasping [right?-ね].
心をつかむ ('to grasp/capture someone's heart') is a fixed metaphor for winning over an audience or person — equivalent to English 'capture hearts.' しっかり is a 擬態語 (state mimetic) meaning 'firmly, tightly, securely,' ubiquitous in everyday speech. ね at the end seeks shared agreement: the speaker assumes the listener also recognizes this. The whole sentence reads as a comment about a celebrity, public figure, or performer.