HRM analysis is performed on double stranded DNA samples. Typically the user will use polymerase chain reaction (PCR) prior to HRM analysis to amplify the DNA region in which their mutation of interest lies. In the sample tube there are now many copies of the DNA region of interest. This region that is amplified is known as the amplicon.
After the PCR process the HRM analysis begins.  The process is simply a precise warming of the amplicon DNA from around 50 ˚C up to around 95 ˚C.  At some point during this process, the melting temperature of the amplicon is reached and the two strands of DNA separate or "melt" apart.
The key to HRM is to monitor this separation of strands in real-time.  This is achieved by using a fluorescent dye.  The dyes that are used for HRM are known as intercalating dyes and have a unique property. They bind specifically to double-stranded DNA and when they are bound they fluoresce brightly. In the absence of double stranded DNA they have nothing to bind to and they only fluoresce at a low level.  
At the beginning of the HRM analysis there is a high level of fluorescence in the sample because of the billions of copies of the amplicon.  But as the sample is heated up and the two strands of the DNA melt apart, presence of double stranded DNA decreases and thus fluorescence is reduced.  The HRM machine has a camera that watches this process by measuring the fluorescence.  The machine then simply plots this data as a graph known as a melt curve, showing the level of fluorescence vs the temperature: