In my search for a simple solution to sleep apnea this is one of the products I have tried and I can confirm that it really does hold the nose passageways open, and encourages more breathing through the nose than the mouth.
The effect is similar to taking a decongestant but without the increased heart rate, and the strips are applied just like sticky plasters. Once attached you only notice them for a few minutes and pretty soon you can't even feel them even if you think about it. In the morning they peel off easily and without pain. When using them air feels cooler since a greater flow is getting through your nostrils, and it's quite pleasant if you have become used to a nose that has become slowly more congested over the years.
However, I use an oxymiter to measure my oxygen levels when sleeping and there is no difference with or without these strips. So if you are thinking of using them for sleep apnea be prepared to find that they may not help. The reason is that the apnea is generally caused by muscles relaxing at the back of the throat, behind both the mouth and nose airways, so they will still decrease airflow even with these strips in use.
Otherwise, to avoid a dry mouth (or dribbling) from constantly breathing through your mouth, these may well be the answer.