Things that intertwine are twisted or mixed together. You have to intertwine yarn to make a scarf.
When things intertwine, they're all mixed up together — it's hard to separate them. To make any type of clothing, threads have to be intertwined. Rope is made of smaller fibers that intertwine. Related jobs—like a writer and an editor—intertwine too, because they are so closely related. You can say people are intertwined if they're close to each other, as family or friends. Getting intertwined could be a metaphor for getting married; the threads of your lives are getting tangled together.