Fingerprints left in the room determined the murderer's guilt.
Literal
Room [in-に] left-behind fingerprints [with-で] that murderer's guilt [subject-が] was-determined.
残された is a passive participle modifying 指紋 — 'fingerprints that were left.' The で is instrumental, marking the evidence as the means of determining guilt. 有罪 and 殺人者 are formal Sino-Japanese legal compounds.