Can I easily modernize the application?
If you have followed the coding structure according to Laravel Docs then you can simply follow the upgrade guides from each version to next. E.g. going from 5.5 guide to 5.6, then 5.6 to 5.7 and so on.
Few months back I gave a similar answer on stackoverflow Upgrading Laravel 5.0 to the latest version (7.x). This should give you a good head start.
Do I need to upgrade?
Not mandatory but highly recommended to do so.
Please read the Laravel Support Policy. In the beginning of this week Laravel 7 has reached end of life. Things are moving fast and in good direction.