Why is earths core molten? What keeps it warm? Where does the energy for the heat come from?

Only the Outer Core is molten, the Inner Core is solid. There are several contributing factors that keep the Outer Core liquid…

The first is primordial heat which is the heat produced by the accretion of the Earth as billions of tiny pieces crashed into one another, their gravitational energy was converted to heat;

The second is that there are radioactive isotopes dissolved in the Core which are continuing to produce energy. There isn't much agreement on which isotopes they are however as uranium and thorium are most common in the Crust rather than at depth;

Then there is energy released by the gradual unmixing of the Earth with the dense Core separating from the rocky material;

There is also an enormous amount of energy being released by the gradual crystallization of the Inner Core;

And finally, you have tidal energy released by the interaction.

The core remains hot because the overlying 2900km or so of solid rock in the Mantle is an incredible insulator.