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
「突込む」 has another reading with different meanings.
バッグに教科書を突込んで家を出た。
I crammed the textbook into my bag and left the house.
話の核心に突込むのが彼女の記者としての強みだ。
Getting to the heart of the matter is her strength as a journalist.
引き出しに書類を突込んだまま忘れていた。
I stuffed the documents in the drawer and forgot about them.