In DSC, differential power (heat) is provided to keep the sample and the reference at the same temperature. The DSC plot has differential power on the y-axis and temperature on the x-axis. Also, as reference is chosen in such a way that it will not undergo any phase change or state change in the chosen temperature range. The instrumentation is not as simple as one might think. There are two separate heaters which can be independently controlled. I quote from O'Reilly's Instrumental Analysis book (slightly outdated but still a pretty good reference for concepts). It describes DSC nicely,
There are two separate heating circuits, the average-heating
controller and the differential heating circuit. In the
average-temperature controller, the temperatures of the sample and
reference are measured and averaged and the heat output of the average
heater is automatically adjusted so that the average temperature of
the sample and reference increases at a linear rate. The
differential-temperature controller monitors the difference in
temperature between the sample and reference and automatically adjusts
the power to either the reference or sample chambers to keep the
temperatures equal. The temperature of the sample is put on the x-axis
(time) of a strip-chart (read "computer" today) recorder and the
difference in power supplied to the two differential heaters is
displayed on the y-axis. The power difference is calibrated in terms
of calories per unit time.
Now imagine that you are heating the polymer and the reference, their temperature is increasing but their difference is zero. You have a flat baseline, as shown in the figure (Taken from Google Images). Let us continue heating both, independently, a temperature is reached when the polymer is melting. There is a phase change and the sample temperature is not changing anymore, but it needs heat to stay at that temperature. The reference, when brought to the melting point, does not need further heating, so its temperature is at the melting point of the sample. The sample is consuming power, but the reference is not. You get a negative peak in the DSC due to an endothermic process.
Apply the same idea to an exothermic process, such as crystallization. The temperature of the sample is becoming higher than the reference. The reference needs power to catch up to the temperature of the sample. The sample heater is not consuming power but the reference's heater is, in order to be at the same temperature. This time you get a positive (exothermic) peak.