Monostable multivibrator with NAND gates
When analysing operation of circuits like these I find it useful to think that the voltage across the capacitor remains constant in the very short term. 


With this in mind, when the left side of the capacitor rises suddenly by 3.3 V (if I read your scope trace correctly) the right side rises by almost the same amount.
The left side is held at 3.3 V by the NAND gate. The right side is now at about 3 V (there are always some losses) has a path to ground through R1 so the voltage decays with the characteristic RC decay curve.
The NAND gate then switches low before the right side of the capacitor has fully discharged* and Vc is about 1.5 V. Since the left side falls suddenly by 3.3 V the right side will try to fall from 1.5 V to -1.8 V (1.5 - 3.3).
D1 clamps the right side of the capacitor at about -0.6 V and quickly provides current to prevent the capacitor going below this level.
We then see a decay from -0.6 V to zero as the remainder of the charge leaks away through the diode. Note that the curve doesn't quite look like a standard RC decay curve. It starts off very steeply just after the falling edge but flattens out a bit too quickly. This is due to the non-linear V-I curve of the diode which means that its "resistance" is increasing as the voltage decays.


* Technically the capacitor is charging at this point as the voltage across it is increasing.

I hope I've addressed the point of confusion adequately. Please ask for clarification in the comments, if required.