Why sign in with a Google Account?
When you set up your phone for the first time, you are asked to sign in with your Google Account. Adding a Google Account is optional, but it's a good idea if you want to take full advantage of the Android platform that powers your phone. Android is an operating system (OS) for mobile devices provided by Google, and a Google Account is tightly linked to the OS.
When you sign in with your Google Account, all your e-mail, contacts, calendar events, and other data associated with that account are synchronized with your phone. Moreover, your data automatically gets backed up to Google servers.
The automatic synchronization means that you can access and edit your data from any computer or device anytime, anywhere. If you make any changes, these will be registered across all your devices where you sign in with your Google Account.
The automatic backup means that all the data associated with your Google Account is stored securely on Google servers, and if you lose your phone or reset it back to factory settings, you can easily restore your previously backed up data.
Moreover, your Google Account username is the same across all of Google products, e.g. Gmail™, Google Play™, YouTube™, Google Maps™, Google+™, etc. This way, adding your Google Account on your phone also gives you easy access to and a smooth experience across all these services.
After first setup, you can add additional Google and other kinds of accounts and sync some or all of their data on your phone. For details, refer to section "Accounts and data synchronization".
Learn more about Google Accounts here: https://support.google.com/accounts/?hl=en#topic=3382296