Hello Customer (name blocked for privacy) and welcome to JustAnswer!
Budgie is the name of a bird, it's short for budgerigar.
http://www.birdhealth.com.au/bird/budgie/budgie-pet3.jpg
Budgies suck on small tough seeds that are hard to open or eat and eventually succeed eventhough they don't have any teeth (actually they do but their teeth are very small and seem useless).
There's a saying "nothing succeeds like a toothless budgie" which means that if you struggle enough you get over any problem. So when they call someone toothless budgie it means a persistent, go-getter person.
Hope this helps,
Have a great day!
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