If furnace duct work is broken, the ducts either have large cracks or part of the duct work is not attached at all. This essentially forms a break in air flow continuity. The hot air your furnace is creating is escaping into your attic or crawlspace instead moving into your house. As a result, your furnace has to work harder to reach the temperatures indicated by your thermostat, and pests have easier access to the inner sections of your house.
If your duct work is broken, the first step is accessing the problem, which means braving either your crawlspace or your attic. Knee pads and protective gear are important if you are planning on making the repairs yourself. You may have to cross insulation and work in tight spaces to find the problem. Working with someone who can shine a light into your ducts from the other side may help, since this makes it easier to spot cracks. Professionals may use smoke machines or other devices to spot broken ducts.
For smaller cracks that occur as part of natural wear and tear, you can typically seal the crack to fix the duct. Liners and thermal insulation tapes (not duct tape) are designed to bind properly to duct-work metal and seal cracks. You can buy these patches at plumbing or hardware stores or hire a contractor to make the repairs for you. New brackets and fasteners can help link up duct work that was not properly connected in the first place.
If the duct work is corroded, you will need to replace sections entirely. Some of these projects require splicing duct-work parts together and may take extra materials and skill, best left to professional HVAC workers. Other projects may involve replacing an entire section of duct work, an easier process that you may be able to tackle yourself if you can identify all the damaged duct work.