Everytime someone performs a search in a search engine, it is a difficult task for search engines like Google to make sure they return the most relevant results possible.
Think of Authority like Popularity. If a search engine has to decide between two sites and which it should rank highest, authority can help. One site may have many links pointing to it from other sites, whilst the other may have little to no links. Search engines can make an educated choice that the first one is the best to place at the top of the organic search results because the popularity and mentions (links) it receives)
The internet is a big place and competition is ever growing so authority is vital to increase web visibility and appearing in as many organic search results as possible.
You can have the most amazing, well built website, but if no one links to you, search engines won’t pay you any interest either.