In a nutshell, mobile first indexing means that one of the largest factors in ranking your website in Google search will now be if it's optimized for mobile devices. Not just responsive. Not backwards compatible. Not "looks pretty good on a phone."
Optimized.
When you open a website in Google Chrome, you can test the site's speed index against Google's own metircs. Here's an article explaining how to open Google's auditing tools in Chrome if you want to try this yourself.
Websites are graded on a scale of 0% to 100%. A perfect score is almost impossible on a modern business website.
A score of 90% or better will CRUSH your competion in speed indexing.
Take a look for yourself: