On violin and viola, for example, scales help students learn the geography of the fingerboard, ie, where all the notes are located. Scales require lots of different finger patterns, and learning these combinations of half steps and whole steps helps students play more in tune. On piano, I find learning scales helps students figure out their way around the keyboard and feel more comfortable playing in different keys. Playing scales with both hands together developed piano students' coordination. As for listening or aural skills, major and minor scales form the basis of most Western music. Learning them helps students recognize patterns of whole steps, half steps, and intervals in every key. I think all young instrumentalists can benefit from learning to sing their scales (ideally using solfege, or at least note names) as well as play them on their instruments--it helps to develop their ears.
Scales are important, but in order to really gain the most benefits from them, we need to learn how to practice them effectively. To often students and teachers drill and kill scales until they're bored of them, instead of finding new challenges to stay engaged. As I mentioned in my blog on the book Make It Stick, students absorb and retain knowledge and skills best when practice is spaced out and and variable. For scales, that might mean practicing different scales everyday, instead of focusing on just one scale at a time. Once my students have learned multiple scales, I try to regularly hear a different scale every lesson. I use books like Galamian's Contemporary Violin Technique, which include tons of different idea for rhythmic and bowing variations. Putting different rhythms or bowings with a scale keeps students challenged and make them put in more effort in their practice.
Do you practice scales or use them in your teaching?