Both use keyword research to uncover popular search terms. The first step for both SEM and SEO is performing keyword research to identify the best keywords to target. The research includes looking at keyword popularity to determine the top keywords or buying keywords that your ideal audience searches for. It also includes looking at keyword competition to see what other brands are targeting the same keywords and determining what you will need to do to compete with those other companies.
Since heading tags typically make text contained in them larger than normal text on the page, this is a visual cue to users that this text is important and could help them understand something about the type of content underneath the heading text. Multiple heading sizes used in order create a hierarchical structure for your content, making it easier for users to navigate through your document.
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.