What muscles am I working on when climbing stairs as an exercise (that's several floors each day)? Do I have to compensate it with another exercise? What effect will be visible after a while?
Most of your lower body will work while climbing stairs but the majority will be done by your glutes. There might be a slight visible gain in size if you don't already go up stairs often it's unlikely though as the resistance in this instance is your body weight. It's almost certain that your glutes have grown to a size required already to move your body weight. If anything you could potentially loose a little bit of weight from it.
This is why back squats are recommended to grow the lower body, you can drastically increase the weight you lower body has to move thus stimulating the muscles to grow.
To complement @lolnotacodecutter's answer:
- No, you do not need to compensate your stair-climbing with another exercise.
- Other than slightly-increased glutes, no other effect will be visible, unless you use this exercise (probably in conjunction with reduced caloric intake) to cause a net caloric deficit such that you lose bodyweight/bodyfat.