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She somehow managed to carry all the bags by herself.
Literal
She [topic-は] somehow herself [with-で] bag entirety [object-を] carried.
自分で (literally 'by oneself, on one's own') uses で in its 'by means of' role to mark the agent as the sole doer — no help received. どうにか ('somehow, by some means') signals that the feat took effort or improvisation. バッグ全部 stacks the loanword バッグ with the noun 全部 ('the whole, all') to mean 'all the bags' — Japanese can append 全部 directly to a noun without a particle. The sentence has the texture of an admiring observation: 'she pulled it off, somehow.'