what is the purpose of adding oil to balsamic vinaigrette slowly?
Specifically, the reason you whisk it in slowly at first is to create small drops. If you just dump the oil in fast, it will adhere to itself and make it impossible to break up into droplets dispersed in the vinegar (or other water based liquid, such as lemon juice). This is the definition of an emulsion: tiny droplets of one liquid evenly dispersed in another. Once you've got the emulsion started, you can pour in the oil a bit faster because the already suspended droplets reduce the room for a big blob of oil to form, and act sort of as "sandpaper" to break up the incoming stream as you whisk.