In the Tachanun prayer, it's traditional to sit for most of the core of Tachanun, but then to stand three words into the last paragraph, between "Va-anachnu lo neida'" and "ma na'aseh" ("And we won't know" and "what we shall do"). Why do we stand up particularly here?
My Rebbi once told me (if I remember correctly) that we do so because that is essentially what we are saying. "We don't know what to do" (vaanachnu lo neida ma na'aseh), so we try everything we can -- laying, sitting, standing.
Aruch HaShulchan (in O.C. 131, 9) posits that now that we have tried the 3 forms of Tefila (as mentioned by yydl and Jeremy) we stand to symbolize "We've tried everything so now what else can we do? Remember your mercy etc."
When Moshe was on Mt Sinai, he received the Torah in three levels: prostrated, sitting, and standing, corresponding to the three levels of tachanun. as for the specific point that we stand, I always had felt it was a transition from the "dejected" part of the tefilla to the uplifted part. i.e., at first we don't know what to do, but with Hashem's help, we will know what to do.