beginning; clue; starting point
Refers to the initial step, foothold, or hint that helps one get started on something. Often used in phrases like 取っ掛かりがない (no clue where to start) or 取っ掛かりをつかむ (get a foothold).
This problem is so difficult that I have no clue where to start.
His one remark became the starting point for solving the case.
Derived from the verb 取っ掛かる (とっかかる), meaning 'to start' or 'to get going', with the nominalizing suffix ～り. The exact historical derivation is uncertain.