。
She said she felt independent through her work.
Literal
She [topic-は] work [by means of-によって] self [subject-が] independent has-become [quotative-と] feels [quotative-と] was-saying.
Two layers of quotation pile up here. The innermost clause 自分が自立している ('self is independent / has become self-reliant') is the content of feeling, marked by と + 感じる. That whole feeling-clause is then the content of saying, marked by another と + 言っていた ('was saying'). によって ('by means of, through') marks 仕事 as the means by which the independence comes about. 自立 ('independence, self-reliance') is a culturally valued trait in Japan, increasingly tied to women's economic agency in modern discourse. Note 自立している as a state, not a process.