猫撫で声 literally means “the voice of a cat being petted”, but we use expression to describe someone asking for a something they want in a very kind, sweet voice - in English, we might translate it as “asking sweetly.”
猫撫で声 is a metaphor from the sound of a cat purring when we pet them. If we pet them and they purr, then this is cat’s way of saying they want more! But once they stop purring, it means that they want us to stop - in fact, sometimes cats may nip or bite at us and jump away! Therefore, 猫撫で声 refers to the sweet, but temporary, voice someone uses to get something from another person.
猫撫で声 is usually used for women and children, likely because male voice is too deep and doesn’t sound ‘cute’ like a cat! 猫撫で声 can also be used as a cute joke, but be aware, it can have a bit of a negative connotation. When you say your friend is using 猫撫で声, this means you can tell they are kind of faking a sweet voice to get what they want - this can sometimes be awkward, so be careful!
子供: 「ねえ、お母さん」 母: 「どうしたの。猫撫で声を出して。」
Kid: “Mom…” Mother: “What’s the problem? You have a very sweet sound in your voice.”
My girlfriend told me she wants a ring for her birthday using her sweet asking voice.