Literally dressing. In this context, it refers to kimono kitsuke. How to wear/dress in a kimono.
Kimono used to be the standard wear in Japan, so everybody knew how to dress in kimono. For formal occasions, special kimonos required some help to look your best (often from a relative), but day-to-day wear everybody did themselves.
To go to kitsuke classes would sound to someone from a hundred years ago like telling someone that you go to school to learn how to tie your shoes. But imagine that shoelaces go out of fashion and for the last decade most people wear velcroed or zippered shoes (who still uses shoelaces .... so old fashioned). But hey you can't were those velcro things in combination with that bespoke dress or suit. Better ask someone to teach you how to tie those damn wires (what are they called again?) correctly or look like a fool.
This is basically what happened after the war. With an economic boom in Japan (we are talking the mid-1960s), there was some renewed interest in kimonos (business opportunity!). People hadn't been wearing kimono on a regular basis for almost a generation, so schools teaching how to dress started popping up. With different styles and their own curriculum, books, shops, etc. etc.
Someone who wears kimono regularly told me once: "Just wear it and enjoy it! There is just one real rule: left-over-right" (the opposite is only used for when you are dead). Again just compare it to day-to-day western wear (a tuxedo or a gala dress might require some rules or etiquette to wear, and you tie your tie a certain way to not stand out too much, (but hey if you dare why not!) but don't tell me how to coordinate my jeans and t-shirts! The same goes for kimono for special occasions there are some etiquette rules but for the rest just enjoy! Only remember LEFT-OVER-RIGHT!
Of course, kimono, nagajuban, and obi are made a certain way to be worn a certain way (just like a tie has a certain length and shape to be tied a certain way). So it comes in handy to know what is the front and what is the back (hard to zipper your pants wearing back to front!).
By the way, the above is not to say that it is not fun to learn how to "properly" dress in kimono, just don't take it too seriously ;)