Most of standing out for most people is either coming up with gigs that are a little different than other people, or coming up with a different angle. I used to have a gig where I would do a write-up for people that read like a newspaper article. There were other people who had the same kind of gig, so I made my own custom template that looked like a real newspaper and although the gig was more about the writing, I put it in the template and it made it unusual.
If there is no way to get a new angle, you’ll have to stand out in some other way - by being better or faster or cheaper than others. You might have to experiment. You cannot guarantee your visibility here. If you offer something really good, though, the buyers will come and with sales and reviews your gigs rise up in search. The search options on the main page is basically the mechanism used on Fiverr.
Some people have to promote their own gigs off-Fiverr to succeed and others don’t. I’m not exactly sure what you mean by promoting your gigs for free. If you mean advertising without paying for it, you can use social media, off-Fiverr forums related to your services, a blog, a portfolio or freebie website or anything else you like. Read some websites or e-books on marketing.