See my Notes! below for a preface!
This is a technique based in misdirection, so most of the examples I've used/suggested/come across follow suit:
1) You hacked into a system to steal or erase a file, and you want to leave as much time as possible so you can dump it or escape before it is noticed. Unfortunately, someone is aware you were having around in the files. You could use Resonance Veil to make them think you were after another file.
2) That security spider hadn't seen you, yet, and you want to make it and it's perception dice go looking elsewhere. Make it think that it saw an intruder somewhere else.
3) Drek, you didn't send the spider off before and now it's noticing that a feed looks different than how it should. Time to make that spider think that the camera is glitching and needs a rest.
4) A runner was trying to incite gang A to attack gang B. Both gangs were pretty level-headed, and though they were rival gangs didn't want to attack each other without a good cause to call the other gangs to. Dropping AR tags of each other's colors wasn't working, so instead they waited until gang A's patrol was around and dropped a tag in front of them, using Resonance Veil to make it seem like it came from gang B. Instant war!
** Notes! **
I'm putting this down here because it's not part of the answer, but might help in understanding the problem that brought forth the question.
First, I feel like I need to address Technomancers: Until Data Trails comes out (soon!), Technomancers have little to go on aside from what's in RAW. A sharp mind can make a basic technomancer that breaks things easily, but fortunately as a GM there are plenty of ways around that. That's beyond the scope of this question.
Second, can this skill be called useless? Sure. It doesn't really have a direct mechanical impact. You can't use it to directly help you with anything, which is why it's a skill better suited for the more covert data wizards.
Third, is this skill broken? Mmmmm....mostly. No more broken than Puppeteer is. And all of these things can be taxing on a technomancer. To make a convincing illusion can put a decent amount of hurt on a technomancer's brain. If you're going against a security spider who has a decent Intuition and Data Processing skill, you're going to want to try a Force 6 to get some net hits. That's five fading you have to resist, and hopefully it will work. If it does work and you hit net hits they now have make a Matrix Perception Test (assuming they feel the need to make one)
Ultimately it's up to you and your GM to make sure it doesn't get too powerful, but if you ask me personally it's just fine. There's a decent amount of payment in the form of fading for the benefit, and the benefit isn't something that will keep your character from dying.