To ebko:

You have to complete certain "quests" to obtain them.

To get the key to Greenwich Square(Working section), you need to give a golden goose to the woman who sells items in the poor section.

To get the key to Hobb Cottage(Hatchery 2), you need to buy it from the woman who is talking to the guy (Rowan), it costs 1400 gold initially. Once you find back Rowan's money for him, she will reduce the price to 980.

To get the key to RoseWood Manor(Hatchery 3), you have to buy it from the woman who sells items in Greenwich Square.

To get the key to East Crest Court(Noble section), you need to give a red gryphon to Rowan (the guy who is talking to a woman).

To get the key to Pemberly Keep(Hatchery 4), you need to buy it from the woman on the right for 120,000 gold in East Crest Court.