This is as simple as it sounds - if the reading was 5mA current when the temperature was 20C, and it changed to 30C, with TCs = 0.01, then the reading could be off by as much as:
$$(0.01 \frac{\%}{K})*(30^\circ C-20^\circ C)*\frac{0.01}{\%}*5mA = 0.005mA$$
While ISO convention dictates the use of kelvin, inverse kelvin are a derived unit and there is no standard when choosing between inverse degrees Celsius and inverse kelvin - except to avoid having to place a unicode degree symbol (kelvin are standalone without the symbol). Another reason for the notation discrepancy is that some measurement devices, such as thermocouples:

have a semi-non-linear response with respect to temperature. As such, these typically will denote their average change over a large range as per degree Celsius, to hint that it is more of an "average" amount and that more precise thermal drift values can be obtained by reviewing the exact temperature range that is being measured.