Fear of separation is something most parents will be familiar with. If it happens to you too, I can only read that as a sign that your relationship to them is working. Yes, it's heart breaking to all of us, but it'd be far worse, I think, if they didn't care whether we left.
You can help them with their anxiety by introducing play into your departure, to lighten up that otherwise sad event. At that age, you will usually get a good response for things like saying "bye bye, I'm leaving" and then take the wrong door, and walk into the toilet or something. "oh no, that wasn't right, now I'll leave", you say, and walk into the wardrobe. I'm not surprised if they'll laughingly show you the door.
Another way would be to put on your clothes in the wrong way, or dress correctly but name the pieces of clothing inaccurately. "I'll just have to put on my shoes" you say, as you put on your gloves, etc.
If it's sad, help make it playful.