Smudge Sticks in Phasmophobia serve two purposes. First, Smudge Sticks can be used to settle an agressive ghost. The effectiveness of this does vary, so don't be too overconfident that you've got some safe time. The second usage is to complete the objective to Smudge the Area near the Ghost.
To use a Smudge Stick it needs to be lit. Don't light this thing too early. It is not like a real smudge stick that can burn for over an hour. This one only lasts seconds, not even a minute.
You should be carrying a lighter and a Smudge Stick. When you have the Smudge Stick selected you can then press F (Default Key) and it will automatically light the Smudge Stick. Once the Smudge Stick is lit, you'll want to walk around the room and then toss it on the ground.
If you are trying to complete a Smudge Stick objective, then you'll want to check the White Board in the truck afterwards. There are two main reasons why Smudging the Area near the Ghost doesn't work. The first is that you've smudged the wrong room. Try another room with another Smudge Stick and see if that will get the objective. The second reason is that these can be a bit buggy at times.
The best advice I can give you with the Smudge Sticks is this: Take two lights on the assignment, in case one gets lost, and take four Smudge Sticks along on the assignment as well. This will buy you a few attempts at smudging. The last tip is that for a $10 objective it's not worth getting someone killed if the Smudging isn't working properly.