Meta title tags are a major factor in helping search engines understand what your page is about, and they are the first impression many people have of your page. Title tags are used in three key places: (1) search engine results pages (SERPs), (2) web browsers, and (3) social networks.
Your title tag determines (with a few exceptions) your display title in SERPs, and is a search visitor's first experience of your site. Even if your site ranks well, a good title can be the make-or-break factor in determining whether or not someone clicks on your link.
Your title tag is also displayed at the top of your web browser and acts as a placeholder, especially for people who have many browser tabs open. Unique and easily recognizable titles with important keywords near the front help ensure that people don't lose track of your content.
Some external websites — especially social networks — will use your title tag to determine what to display when you share that page. Here's a screenshot from Facebook, for example:
Keep in mind that some social networks (including Facebook and Twitter) have their own meta tags, allowing you to specify titles that differ from your main title tag. This can allow you to optimize for each network, and provide longer titles when/where they might be beneficial.