As a first approximation, it's all down to air density.
A quick back of envelope calc tells me that in going from 15°C (ISA Sea-level temperature), to 25°C (ISA+10°C), at the same pressure equates to a drop in air density of about 4%, or equivalent to being about 1200ft higher airfield elevation.
I had a hand in producing a ready-reckoner for GASCo a little while ago which helps you work out the effects of elevated temperature on a light aircraft. You can download it in a handy kneeboard format
here, or there's an article about it
here. As you'll see from there, that increase in temperature from 15°C to 25°C should give you about a 20% increase in TODR.
As you say, not unlike taking off with Carb Heat on - since basically the same is happening, air density is dropping with the increased temperature.
G