If the problem with the device is only those clearly damaged components, I think you can give it a shot. If there's something else wrong (which caused that resistor to slightly explode in the first place), it would at least be beyond my paygrade, I wouldn't recommend spending time trying to fix it if you're a complete novice. If my guess is correct, the problem is indeed the overheated resistor and the blown thermal fuse and it should be possible to fix-ish.
Yellow box is definitely a capacitor (a radial film 0.1 μF 275VAC X2 class capacitor to be specific - typically used for EMI suppression). It's possible that the board will work without it (as seems it's only used for filtering/interference suppression) but I would replace it. If you do replace it, definitely spring for an X2 class as well, don't cheap out or you can end up with a spicy explosion. You will need the same capacitance (0.1 μF / microfarad) and the same or greater voltage (at least 275 VAC / Volts on Alternating Current). Maybe even get the same one - https://www.ebay.com/itm/405324404548 although $10 for just two capacitors is insanely expensive. Here are some more which should work: https://eu.mouser.com/c/passive-components/capacitors/safety-capacitors/?capacitance=0.1+uF&safety+rating=X2&voltage+rating+ac=275+VAC (just make sure to get the Lead Spacing right - I'd guess yours is 10 mm, but better to measure out).
The device below that blown resistor is likely a thermal fuse, designed to prevent exactly the situation that happened (that means two things - this exact event has probably happened during manufacturer testing, and your thermal fuse is not up to spec). I would try replacing it with a thermal fuse that's at least 240VAC rated voltage, something like 10 A rated current, and something like 150°C trip temperature.
It's a bit hard to identify the exact value of the resistor because it has discolored so much. It's especially annoying since the third band kinda has the biggest effect on the value (it's the exponent) and that's gone completely. I would try desoldering it, measuring it with a multimeter (on Ω/Ohm setting) and see what value comes out, then get a 2W or 3W resistor of a larger value and see what happens. This is the most risky part of the whole ordeal, if you get it too high the thing will probably not work, if you get it too low it might explode again. Probably would stay away when plugging it in for the first time :)
Good luck, and be careful. In any case it would probably be a fun new experience - getting components, soldering them, and then either you get a working blanket or some tiny very dangerous fireworks :)