Domain authority is an important ranking phenomenon in Google. Nobody knows exactly how Google calculates, ranks and rates the popularity, reputation, intent or trust of a website, outside of Google, but when I write about domain authority I am generally thinking of sites that are popular, reputable and trusted e.g. are also cited by popular, reputable and trusted sites.
However, that’s totally impractical for established sites with hundreds of pages, so you’ll need a tool to do it for you. For example, with SEMRush, you can type your domain into the search box, wait for the report to run, and see the top organic keywords you are ranking for. Or, use their keyword position tracking tool to track the exact keywords you’re trying to rank for.
Ask for a technical and search audit for your site to learn what they think needs to be done, why, and what the expected outcome should be. You'll probably have to pay for this. You will probably have to give them read-only access to your site on Search Console. (At this stage, don't grant them write access.) Your prospective SEO should be able to give you realistic estimates of improvement, and an estimate of the work involved. If they guarantee you that their changes will give you first place in search results, find someone else.