The shifters look like Tiagra.
Unhitch the cable from the brake stirrup. Pull the lever and you should see the 'pear' nipple in an aluminium barrel with some bushes on each end. with a thin screwdriver, pick the barrel out of its seating and tease it out of the front while gently pushing the cable at the brake stirrup end.
Watch the bushes, they will fall everywhere, so dab them with grease to make them stay on the barrel.
Pull the cable fully out.
Before you install the new cable, coat it with waxoil. Make sure the snipped end is correctly wound and put a dab of grease on it. Put the barrel on it. Note which way the Bowden cable is wound and ONLY rotate it so it tightens. You DO NOT want the Bowden cable fraying inside the outer. GENTLY thread the cable down into the shifter. The idea of using a torch is GOOD. If you hit a stop, don't force it. Rotate the cable to tighten it and it will prob' slip into place.
When the cable is fully installed, clean off the waxoil from where it will be clamped with meths on a kitchen towel.
Good luck.
Thank goodness I didn't attempt a BB service.