What causes hot things to glow, and at what temperature?
It is due to thermal radiation. Bodies with temperature above absolute zero emits radiation. If frequency of the radiation is in the visual range the body "glows".
When the electrons in the atom are excited, for example by being heated, the additional energy pushes the electrons to higher energy orbits. When the electrons fall back down and leave the excited state, energy is re-emitted in the form of a photon. -wikipedia(emission spectrum)
This explains that heat itself is not giving off the light.
Spectral energy density as function of of wavelength and temperature
$u(\lambda,T) = \frac{8\pi hc}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k T}}-1}$
If you integrate $u(\lambda,T)$ wrt $\lambda$ from $\lambda = 380nm$ to $\lambda = 750nm$ you will find that there is always some radiation in visual range. But it is very very small at room temperature(T = 300K) and hence undetectable to naked eye.
To get cut off temperature($T_c$) for glow. You need to solve
$\displaystyle \int_0^{T_c} \int_{\lambda = 380nm}^{\lambda=780nm} u(\lambda,T)d\lambda dT$ = minimum required to be detected by human eye.
I would like to emphasize a point mentioned in the first answer.
The classical black body radiation has to do with a gas of photons in thermal equilibrium. The photons interact with each other and that interaction gives the usual Planck spectrum for their energies.
Inside stars, there are photons produced which interact with the stellar matter and they both come to a thermal equilibrium. When the photons escape the star, they have a black body spectrum.
When you take some amount of matter (an electric stove) and heat it, there is the question of where do the photons come from. In low temperatures, you don't have enough thermal energy to excite electrons in atoms. So, where do the photons come from? The answer is that they must come from the excitation of the background electromagnetic field. Even if you tried to isolate the heated matter from every electromagnetic field, you could never block out the vacuum fluctuations.