The same way you can have vegan sausages.
Vegetarian haggis is made with all the vegetable ingredients in regular haggis, plus lentils, beans and other veg. It looks like a haggis, uses the same spices as a haggis, tastes somewhat like a haggis, and just doesn't have any of the animal bits.
'Proper' haggis is made from sheep's heart, lung and liver, minced and mixed with onions and other stuff and then stuffed into a sheep's stomach. Basically, it's kind of like a sausage, which is meat of some kind (quite often bits not used for anything else), spices and sometimes other vegetable stuff, traditionally stuffed into an animal's intestines. Much the same thing.
...For someone who's never eaten haggis I certainly know a lot about haggis, huh?