Good question! The structure of diamond is a 3-dimensional lattice of carbon atoms strongly (covalently) bonded, each to four other carbon atoms. The strong covalent bonding means that when you heat the material the thermal vibrations are rapidly transmitted through diamond. On the other hand, the lack of free electrons in the structure means that, unlike graphite, it is a very poor conductor of electricity.
My chemistry is a bit rusty, but consider the crystal structure of diamond. Each carbon atom makes four bonds with four carbon atoms around it, arranged in a tetrahedral fashion. These bonds (by which two atoms share two electrons each) are very strong, which is why diamond is so hard. It also means that, since all electrons participate in such a covalent bond, there are no free electrons to move around, which means no (or very low) electrical conductivity. The particular crystal structure of diamond is also the reason for its high thermal conductivity, but I have to admit, I don’t know enough about thermal conductivity to venture a guess as to why that is.
The molecular structure of diamond is very rigid and uniform which is perfect for thermal conduction (the thermal radiation moves along the molecules easily). However it doesn’t have any spare electrons which is what is necessary for electrical conductance, hence it insulates.