Electric baseboard heat question

Hello. I'm a new homeowner and I'm trying to unlock the secrets of my
house In my basement we have electric baseboard heaters, 240V.
Curious thing is, the thermostat that controls them is a 24V model!

I traced the wires coming from the thermostat to a box attached to the
side of my breaker panel. It's a Honeywell box, but that's all I
could tell from the outside.

I didn't want to mess with 240V too much, and I haven't had the
opportunity to cut the mains and remove the cover, but I did remove
the cover enough to see what looks like a beafy relay or contactor or
something along those lines.

So, my question is this... what is that box in all probability? Is it
really just a relay box in essence? If so, can someone point to a web
site that might fill in some of the details for me?

I'm interested in installing those fancy LCD touchscreen thermostats
everywhere, and something like this is obviously what I'll need, so
I'd like to understand how this is hooked up and what it is
(interestingly, nowhere else in the house has this, everywhere else is
standard line voltage thermostats, all electric heating throughout the
house).

Thanks all!