A bad recessed lighting shortcut

Electrical Question:
I am an HGTV buff. I have been watching income property for some time now.

They've been saying there is an illegal short cut that some contractors use when inserting recess lighting. Where as the ceiling next to the light is hot.

When that side of the drywall/and or ceiling is removed, there is this box. Is that called a Bot Box as I thought I heard them say it was called or is it called something else?

It resembles a junction box, but if it is, why is it hot which can cause fires?

What HGTV may be referring to is that most locations require recessed lighting that is installed into an Insulated Ceiling to be a IC Rated fixture enclosure, and in some areas it is required to be an Air-Tight or AT Type enclosure for energy efficiency purposes. There are heat factors with regard to proximity of insulation materials and in some cases it is necessary to install a barrier or a recessed light that has a box enclosure around the recessed light can.

Sometimes this is referred to as a Boot Box and an inspector may require this if the recessed light fixtures are not an IC type enclosure. I have seen scorched cellulose insulation that was too close to recessed lighting that was not IC rated.
Be Careful and Be Safe - Never Work on Energized Circuits!
Dave Rongey