To cling to or hang on to something (physically); to hold on tenaciously.
子猫が母猫の首に食い下がって離れなかった。
The kitten clung to the mother cat's neck and wouldn't let go.
崖でロープにしっかり食い下がる必要がある。
You need to hold onto the rope firmly on the cliff.
To hound, persistently pursue or refuse to back down from someone; to press on tenaciously (in argument or negotiation).
記者は答えを得るまで食い下がって質問し続けた。
The reporter kept pressing the question until he got an answer.
彼は相手の言い訳に食い下がり、真相を暴いた。
He refused to let the other's excuses stand and exposed the truth.
Sumo term: to grab the opponent's front mawashi, press one's head to their chest, and lower the hips to gain advantage.
力士は土俵際で相手の前まわしを食い下がり、形勢を逆転した。
At the edge of the ring, the wrestler grabbed the opponent's front mawashi and overturned the situation.