Natural Selection Naturally

In my mind there is little doubt that species change. If you take a look at birds, there are birds that are highly similar but have different beaks or claws. It is probable that these birds came from the same common ancestor. Darwin proposed that the cause of this is natural selection. Within a single species is a incredible variation in physical attributes. Just look at human beings. If the environment that a species is living in gives an advantage to a particular attribute, the beings that have that attribute would have a higher chance of survival and therefore should increase in numbers compared to the beings that do not. Take for example the industrial revolution in the United Kingdom. Many factories spewed a great deal of soot into the air which coated neighbouring trees and made them darker. Moths that were darker in colour had an advantage over moths that were lighter in colour because they were less visible to their prey on the darkened trees. Therefore, there was an increase in the number of dark moths compared to their lighter cousins. Competition for food can also result in natural selection. If weather patterns create a more aquatic environment where fish are deeper in the water, those birds with longer beaks will have an advantage over those with shorter beaks and, over the long term, should become more numerous all other factors being equal. This makes sense to me. What I do not understand is what it has to do with evolution. In my mind they are two different subjects. Natural selection is nature picking out certain traits. It is like nature's dog breeding program. It is like taking 100 000 dice that are in 50 different colours and rolling them, then having five seconds to pick out all the ones you see that are pewter grey. If you do this a few times, you will have less of the pewter grey. This is not an increase in complexity. In my mind, this does nothing to create a more complex organism. I don't understand why natural selection and evolution are often tied together. To me they seem like two separate ideas.