Energy and matter are equivalent. They can be considered as one single thing, mass-energy.
The following is an analogy. I'm going to ignore for now the true fact that adding energy also adds mass during this analogy.
Let's say you had some ice. There's a certain number of water molecules in the ice.
If you warm the ice so it melts, keeping all the molecules, you will have liquid water. If you only melt part of it, part of it will be liquid water, and part will be solid ice. But you will still have the same mass of water. It's just that some of it will be solid, and some will be liquid. But you still have the same amount of water. The liquid water has properties different from the solid water, but it's still all water. The ice is solid water, and the liquid water is liquid.
Now, if you warmed the ice in a sieve, the liquid water would drain away, and you would have less total water in the sieve. That drained water would still exist. It might go on to be absorbed by a plant, and have the hydrogen in it turn into a sugar, and the oxygen released to the air, or any number of other things. But the mass of that water molecule would still exist, perhaps split up into many different places.
In the same way, if you have matter in the Sun, and it fuses, giving off energy, you have changed the properties of that portion of the Sun's matter that was converted into energy. Matter turned into energy has different properties from matter, just as ice melted into water has different properties from ice.
But it still exists. And just as the sum of ice and melt water adds up to the same amount of water as was in the original block of ice, the sum of energy released and remaining matter released by fission, fusion or matter-antimatter annihilation remains the same. Some of the energy might warm the Earth, some might warm the moon. Some go on and add a teeny bit of warmth to planets around other suns.
So far as I know, there isn't any way to create matter or energy. It can only be changed into a different form.
So the Sun converts part of its matter into energy in the form of photons, and photons, having different properties than matter, will wander off into the universe, just as liquid water, having different properties from ice, will drain through a sieve. So the Sun loses mass, just as the melting block of ice loses mass. It's not that the mass the Sun had doesn't exist anymore, it's just that it's somewhere else, doing other things.
To address the part I ignored for the sake of simplifying the analogy, it shouldn't surprise anyone that there is more mass-energy in the sum of ice and meltwater than there is in just the ice. You added mass-energy to the mass-energy of the ice to melt it into water. Of course it's larger.
