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A linear regulator makes a clean, steady voltage by burning off the extra as heat, and that heat can cook the part if there is too much. This works out how much power it dissipates, how hot its junction gets in your package, and whether it needs a heatsink, from the input and output voltages, the current and the cooling.
The drop becomes heat
A linear regulator does not store or convert the surplus voltage, it simply drops it while passing the full load current, and that dropped voltage times the current is power turned to heat. Feed 12 volts in for a 5 volt output at half an amp and it burns three and a half watts, every bit of it as heat in a small package. The bigger the gap between input and output, and the more current, the more heat, which is why a linear regulator from a high rail runs hot.
From watts to a junction temperature
That heat raises the temperature of the silicon inside, the junction, above the surrounding air. How far depends on the package and cooling, given as a thermal resistance in degrees per watt. A bare TO-220 in still air is around 60 degrees per watt, so three and a half watts lifts it nearly 220 degrees over ambient, far past safe. A heatsink drops that resistance to single digits. Junction temperature is ambient plus power times the thermal resistance, and it must stay under the part's limit.
When you need a heatsink
If the junction lands above the limit, usually about 125 degrees, the regulator will throttle, shut down or fail, so you cool it. A heatsink is the direct fix, cutting the thermal resistance so the same watts raise the temperature far less. The calculator shows the junction temperature for your package and the most current you can draw before it hits the limit, so you can see at a glance whether a bare part will do or a heatsink is required.
Or stop making the heat
The other answer is to make less heat in the first place. Narrow the gap between input and output, since the drop is what becomes heat, so regulating 6 volts to 5 wastes far less than 12 to 5 at the same current. Or use a switching regulator, a buck converter, which chops and stores energy rather than burning the surplus and runs at eighty to ninety percent efficient, barely warm. For anything more than a small drop at low current, switching is usually the better choice.
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