There are two main reasons why gaining inbound links to your website is important: links are a ranking factor and they generate traffic from both search engines and third party websites.
You only have to read Google own developer blog titled "Good times with inbound links"
to find information on this.
If you don't believe Google , as many don't, you can refer to 3rd party trusted sources such as MOZ's "search ranking factors" page . This outlines how Google understands how trustworthy a webpage is based on the number and quality of inbound links it may have directed to it.
Why? Because websites tend not to direct people away from their own site without good reason. Plus, and most importantly, Google knows that it's out of the hands of SEO's & website owners. They can't force quality websites to add their links, so Google has developed their algorithm to learn which sites to trust based on their content and link signals e.g. it's bad if a website links out to too many sites without many inbound links (spammy).
Having links from other websites also boosts the volume of traffic coming to your site outside of Google (and other search engines). Depending on where your customer are in the buying process, they may be using other sites - such as local directories, industry publishers, influential bloggers etc. to find answers to their queries. Being listed on resource pages listing the top businesses in your area for example, means that your customers can find you through multiple avenues.
But as links are so important it's no surprise that they can be difficult to obtain. It used to be the case that some SEO companies would attempt to 'game the system' by using 'black hat' link building tactics that no longer work. Search engines have got wise to these shady tactics, such as buying links, submitting to spurious directories, and spamming blog comment sections and online forums with links back to your site.
But this doesn't mean that you can't find ways of improving the quality and quantity of links pointing to your site...