I won't leave home without them!
Advantages of trekking poles:
Improve stability on rough areas, stream fording, soft snow. (On steep or icy snow, though, an ice axe is preferable
if you know how to use it for self-arrest.)
Improve balance (great for us older folk who tend to lose it as we age)--they have saved me from a number of potentially serious falls.
Take stress off knees, ankles, feet, hips, especially on downhill. Help prevent stress injuries for those whose joints are still pain-free, and really help those of us with old injuries or arthritis.
Less leg stress on the uphill because arms and shoulders help with the effort, also increase endurance.
Easier to maintain a good pace on level ground, at least for me, again because the arms and shoulders help.
At home, when exercise walking, poles turn walking into a whole body exercise (Nordic walking) that works those important core muscles.
Multiple uses:
Shelter support (save weight of tent poles and also sturdier)
Travois foundation (in case my dog gets hurt)
Splints
Hand crutches
Fishing rod
Chair support (prop up my pack to use as a chair back)
Ski poles (with snow baskets added)
Fend off loose dogs
Wave at mountain lions (should I meet one) to make me look bigger
Potential weapon against 2-legged varmints
Convenient measuring tool, especially in photos (be sure the centimeter markings are towards the camera)
Trail maintenance tool (recommended by the USFS)--push sticks or stones off the trail; clear drainage ditches. You can do these as you hike, generally without breaking stride.
Cons:
Get in the way on brush-choked trails and when scrambling. If you have collapsible poles, they can easily be collapsed and fastened to your pack for such sections.
Keep your hands full. However, I have learned both to handle my dog's leash and to get my camera out and take photos without dropping the trekking poles. In the case of the leash, I can do this while moving. It just takes practice and (for the dog) some training.
Poor technique can make the poles ineffective.
Here's everything you ever wanted to know about trekking poles.