thorough investigation; complete inquiry
Used for exhaustive probing or getting to the bottom of a matter, often in formal or serious contexts such as accidents, scandals, or research.
We need to conduct a thorough investigation into the cause of the accident.
The committee promised a complete inquiry into the issue.
究明 means 'investigation' or 'inquiry' in general, while 徹底究明 adds the nuance of doing so exhaustively and without leaving any stone unturned.
Compound of 徹底 (thorough, consistent) and 究明 (investigation, inquiry). The exact historical derivation is uncertain, but the meaning is transparent from its components.