Summarize:

This paste is heated up and creates a strong electrical connections between the two bits. Apply the paste liberally with a cotton swab. Some of the flux will squeeze out, but this is okay. Get the lug snugly onto the exposed cable for now. If you have a crimping machine, now is the time to use it. Otherwise, get a flat nail punch or angle iron, and a hammer. Place the punch in the center of the nut, then hammer hard into it to crimp the terminal into the cabling. Flip the cable and repeat on the other side. You want to get the entire piece nice and hot. You can stand a butane torch up, turn it on, and rotate the cable in front of the flame so the whole thing is evenly heated. Heat until the rosin starts to bubble out of the terminal lug. This preheats everything to make soldering easier in the next step. Use your non-acidic, non-reactive solder wire to solder the point where the terminal lug ends and the exposed cable begins. You can do this right after you have the rosin boiling. When done, let it cool for 1-2 minutes before proceeding. Make sure you have enough heat to liquefy the solder. All that needs to be exposed is the flat terminal lug, which connects to the battery. The rest of the connection should be covered and snugly fit with the sheathing to prevent corrosion.
Liberally apply flux paste to the inside of the terminals and the exposed cables. Slide the terminal lug onto the expose cable end, pushing it as far on as you can. Crimp the terminal lug onto the cable using a flat nail punch or a crimper. Using a flame or other high-heat source (such as from a butane torch), heat the lug until the rosin flux is boiling. Still applying heat, solder around the connection between cable and terminal lug. Slide the shrink tubing back up to cover the expose metal, then use the heat source to shrink it around the connection.