Someone knocks at the front door. One of the family calls out to the others, "I'll get it", so that no other member of the household stops what they are doing and goes to answer the door unnecessarily.
Someone knocks at the front door. The mother calls out, "Frank, can you get that? I'm in the middle of changing the baby." However, Paul is just coming down the stairs and the front door is close to the bottom of the stairs; so Paul calls out, "I've got it." This is to indicate to Frank not to bother; and to the mother, indicates "I've got it instead." At the same time, Paul could just have well have said, "I'll get it."; "I've got it" implies more that he is close to the door and is just about to answer the knock.