There is no 突っ込む or conjugations in the text shown; it's all in hiragana and katakana - but this will come up as an IME suggestion, so it's worth addressing. The text is not very clear, but I agree it must be つっこまれ。。。 which you may have instead read as つっこまない.
The original sense of 突っ込む is to thrust into something, stuff something, etc. Basically, to enter a space forcefully. [突]{つく} is to stab, poke or thrust. 込む by itself is something like "to be crowded", but in a compound like this it basically does the same thing that "in" or "into" does in English phrasal verbs like "thrust into", "jump in" ([飛]{と}び込む), etc.
More metaphorically, one might 突っ込む into some complex matter - in a more natural English metaphor, get to the heart of it.
Even more metaphorically, we get the sense from Japanese comedy, at which point we normally start writing the word in katakana. The ツッコミ is the one who plays a role which requires つっこむ, which is to say, cutting right to the point of what is wrong with what the ボケ just said or did. (Grammatically, つっこみ is of course the i-stem of つっこむ; and i-stems often have a noun or noun-like function, although the exact relationship to the corresponding verb varies.)
This form of comedy needs to be watched to be properly appreciated; describing it dryly in English doesn't do justice to how funny it can be. (Disclaimer: the following is my own poor attempt to summarize my own limited experience with the form.) Sometimes people translate by saying things like "playing the straight man", but the typical Japanese act works too differently for that to make a lot of sense. In Western comedy, the "straight man" talks/acts first, to set up a scene where the "joker" can take the audience off guard (whether by wordplay or by simply acting the fool). But generally in the boke-and-tsukkomi routine, the boke first does something "off" that the tsukkomi can point out to deliver the punchline, in "what's the deal with airline food anyway?" fashion. The boke might still do ダジャレ wordplay, but quips - an acerbic tongue - are the tsukkomi's responsiblity. (The simplest and plainest such retort is simply to react to some ダジャレ with 「寒い」.)
As sundowner notes, the woman's internal thought process in this case itself involves wordplay; and the sexual interpretation should also be obvious from the original sense.
