As others have said, looks like rust/corrosion increasing friction.
To repair teporarily You could try dripping some solvent and lubricant onto the inner cable so it weeps up into the outer. This might get it moving slightly better, but is not really a fix.
To Repair PROPERLY: replace the inner cable completely, and depending on the condition the outer sheathe too. This is not too hard and can be done by the home mechanic.
You'll need a replacement inner cable with the right mushroom head. Look inside your brake lever to see what the end looks like. I'd be 99% sure its a MTB or off-road design, which is a cylinder shape with the wire coming out the curved side (right in image). The road one is a double cylinder with the wire coming out the centerline vertically. (left in image)
You simply undo the retaining bolt/nut at the brake, feed the wire out, undo from brake lever, and then Haynes it back in.
If the new cable is hard to get through the old outer, you might have corrosion or wear in there too.
A kit with enough to do a whole bike is less then the cost of lunch, and will contain two inners and enough housing to do the lot. Example:
As for tools a cable cutter would be nice, but not required. Trimming the excess inner can be hard - check the answer at How to cut a brake/derailleur cable? for tips and techniques there.
Prevention I can't see in your video or image, but rust is caused by water and aggravated by salt. So keep your bike dry and wash off any road salt after wet rides. Also helps if your bike is stored in the warm and dry overnight. Outside in the rain will certainly accelerate rust.
It is possible your rear brake is poorly routed too. If the lowest point is not an end then water will pool inside the cable. If this is the case consider rerouting the cable on the frame, perhaps up the chainstay. You can also use longer lengths of outer housing and skip some of the frame stops. Water can't get through the outer housing, it only goes in the ends.
Another thought is to protect the end of the cable better. You could possibly put a squirt of marine grease up the outer before fitting the inner. This should provide some level of seal to reduce water ingress, but it reduces water egress too.
Last thought, a V brake noodle-boot may provide additional coverage if threadded over the inner, where it exits the outer down by the caliper. I'm talking about the black rubbery bit here - (its only half the cost of lunch)
All this is within reach of the home spanner wielder. Any questions do ask here or check in [chat]