Why for an inductor part is the RMS current rating is specified at 20°C rise and 40°C rise?
At those ratings the component's temperature rises 20&deg;C resp. 40&deg;C with respect to the ambient temperature. So, at an ambient temperature of 35&deg;C  the inductor's temperature becomes about 55&deg;C resp. 75&deg;C.  

Footnote 6 says these values are tested/measured at an ambient temperature 25&deg;C. So, at other ambient temperatures, the temperature rise is an indication / reference value.

NB. Ambient temperature  means the inductor's ambient temperature . So, if it is outside 35&deg;C but your electronics is in a black box in the full sun, the ambient temperature for the inductor might be 75&deg;C.  

Additional andwer to additional question
Normally, you don't consider or pick a temperature rise and try stay within the rated current. It is the other way around: You calculate your rms current and calculate the rise in temperature (or use the derating link in the datasheet) and make sure to stay within the thermal limits.