Why is a kilogram of feathers and a kilogram of lead the same amount of weight?

In a vacuum, both of these weigh exactly the same - 1 kilogram. Because kilos are a measure of weight, and not density, any substance that weighs 1 kilo will weigh the same - 1 kilo.

The term that helps us understand the difference between the lead and feathers situation is Density. The density of an object is a measure of its Volume and Mass.