Vegetarian diet isn't really connected to low-fat. Meat generally decreases your food's average fat content, since most meat is lean. Vegetarians make up lack of lean meat with more fatty dairy and cholesterol raising eggs.
It's my understanding that dietary cholesterol consumption is not correlated with blood cholesterol levels.
Except a ton of vegetarians don't consume dairy or eggs because they're made from animals.
those are vegans.
There are also non-vegan vegetarians who eat a lot of vegan food, or are health or environmental conscious so they minimize egg/dairy while not cutting it out completely.