The reason you should eat more eggs to lose weight is that they are rich in things you need for health, low in overall calories, and will keep you feeling full for a long time, making you less likely to overeat on snacks between meals.
I particularly enjoy eating an egg-based meal for breakfast, as I feel like this serves me well later in the day. My lunch is likely to be more appropriately sized if I eat a hearty breakfast.
I enjoy an egg and banana flapjack, scrambled eggs or an egg-laden quiche. A great dinner is a giant (5 egg) omelette packed with cheese and veggies.
The bottom line is that eating more eggs should help you displace some less healthy foods from your diet. They are healthy and cost-effective, and very tasty.
Eat more eggs, lose more weight.