All birds have a distinct way in which they sing or chirp. Once one can identify the kind of sounds a particular bird makes, they will be able to identify or realize the presence of that bird by that same sound. Our elders observed that this sound does not change and likened this to habits within humans. Once one gets used to doing some things in a certain way, they seldom change. Hence it is considered second nature to them and it is of no use trying to change those habits in those people.
The proverb is used to comment on characteristics (usually, but not limited to, bad habits) that are constant in someone but as a resignation to that fact that those characteristics are there to stay. It can also be used when people are fed up with the way someone behaves and they comfort themselves by pointing to the fact that this is just the way that person is.