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
棚の奥に荷物を無理やり突っこんだ。
I shoved the luggage forcefully into the back of the shelf.
彼の説明の矛盾点に突っこんで質問した。
I pointed out a contradiction in his explanation and asked about it.
ポケットに手を突っこんだまま歩いている。
He's walking with his hands shoved in his pockets.