It's an old SEO trick. You analyze your competitors backlinks using a free tool or a paid tool like ahref. You then go to each web page, and run a broken link analysis (there are many different Chrome plugins for this, search for broken links extension).
Then contact the page owner and kindly let them know that one of their links are not working anymore. Once they respond to thank you for helping them out, respond back and let them know that your website could be a relevant link to add on the same page they added your competitor.
Because you helped them out, they will feel much more inclined to add your link. It won't work all the time, but it dramatically increases the chance of them responding.
It's a really old SEO technique but I felt I should share in case someone haven't tried it out yet.