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"Gauche" means "lacking ease or grace; unsophisticated and socially awkward." An offer of sympathy is condolences. As a hypothetical example, if someone ran over your cat and said, "I'm so sorry! Let me buy you a new one!" That would be a gauche offer of sympathy. Bumbling means clumsy, awkward. So the sentence is saying "a gauche offer of sympathy"(unsophisticated and socially awkward offer of sympathy) is awkward and clumsy as to be embarrassing. Embarrassing: Something that is embarrassing makes you feel shy or ashamed

Basically it's saying an overly-obvious and showy display of sympathy makes everyone feel awkward and embarrassed.