When the thermometer's in the patient's mouth (or wherever), it expands up to the temperature of the patient.
As it's pulled out, it starts to cool and contract back into the bulb.
The kink is designed to 'break' the thread of the mercury (or alcohol), so the part closest to the bulb contracts back into the bulb, while the remaining section of mercury just contracts where it is in the tube.
This means that the temperature read off the scale is a lot more accurate.
Without the kink, more liquid would contract back to the bulb, leading to consistently low readings.
To reset the thermometer, you give it a quick shake so the liquid's all together again.