Can I use songs downloaded from Apple Music in a home movie?

Yes and no. Songs downloaded from Apple Music are copyrighted and grant the producer or artist exclusive rights for their use and distribution. However, there is something called the "Fair Use" doctrine in copyright law that allows certain use without getting permission. So, if you are making a home movie that only you and your close knit family are going to see and you don't plan on selling the movie or using the movie for promotional purposes, I doubt that Taylor Swift is going to come after you and sue you. But keep in mind that she could if she wanted to. My guess is she would consider it a compliment that you chose her song as background music for your family's joy and entertainment.