I would recommend this introduction to type theory: https://arxiv.org/abs/1601.05035 https://arxiv.org/abs/1601.05035. While it's about HoTT, it starts with a

I would recommend this introduction to type theory: https://arxiv.org/abs/1601.05035 https://arxiv.org/abs/1601.05035. While it's about HoTT, it starts with a very clear general-audience explanation on why and how type theory is different from set theory.