Because you have separated the plants from the predators that were keeping the aphids down outside.

Mostly people get rid of aphids by physical means, a blast of water spray to knock them off (do this outside obviously), and squashing them with their fingers. There are also "natural" sprays which keep aphids down, based on plant oils such as rapeseed oil and neem oil; again you would need to apply them out of doors, which I have used and find reasonably effective.

Googling "aphid houseplant" will no doubt give you some other ideas. Obviously if you want to eat next year's pepper crop, systemic insecticides are a bad idea.