3 wires hanging from fixture
Firstly, some safety points/observations on your content...


Never, EVER, EVER, connect wires if you do not know the original connections without some pen &amp; paper research, testing, and writing down what goes to what
If wires have burned, there are safety concerns to observe. Did the wires burn at the wire nuts ONLY (due to high resistance), or is the circuit overloaded and the wires melted due to too much current being pulled over an overloaded circuit? Again, this is paper and pencil work. What wire type is it? Looks like a standard copper 15 Amp circuit. Ensure the breaker is 15 Amps not larger, add up the demand on this circuit, are you trying to get more than 15 Amps from it?
Has the breaker been tripping often? How many amps is the space heater?
Due to the fact you've noted the wires are burned, make a list of all of the things on this circuit, and check the wires where the heater was plugged in, here at the fixture, and any points up to the panel, including looking at the wires for this circuit INSIDE the panel. If they are crispy and melted/burnt, you will need to replace the run
I'm glad you pointed out the sheathing missing on the wire, you need to check if the sheathing is crispy or broken elsewhere, all the way back to each of its connections. Do not use the wiring if it is. If it is only here, and CONFIRMED only here, you'll need to tape it/replace it.
If this is daunting, it would be best to hire an electrician to come in to do the repair, safety first!!


Most likely yes, the two blacks belong tied together (the whites probably do too.... but this all needs to be validated), but since the circuit blew, and the light no longer works, we know something in the circuit is not working. The burned wires need to be replaced. This is an indication that some other weak spot (at additional wire nuts, or the panel, or the breaker, and it could be hot or nuetral side) is broken/dangerous. You really do need to check it all the way back. You really do need to make sure you aren't exceeding the circuits capacity. Also, ensure the breaker is the right size. My post here is somewhat redudant, but it is very important to do these checks.