Fetching details from the dictionary...

to thrust (something) into (something); to cram; to stuff; to shove
to plunge into; to charge into; to rush into; to ram into; to crash into
to delve into (a matter); to go into depth; to get to the heart (of something)
to press (someone) about; to point out (e.g. an inconsistency); to question sharply; to grill
to get involved with; to meddle in; to poke one's nose into
to riposte; to retort; to quip
ポケットに手を突っ込んだまま歩いている。
Walking with hands shoved in pockets.
車がガードレールに突っ込んで事故が起きた。
An accident occurred when a car crashed into the guardrail.
彼は問題の核心に突っ込んでいく姿勢が素晴らしい。
His attitude of getting to the heart of the matter is wonderful.