I can manage my team on a daily basis without interrupting their workflow. We can keep track of each other and tag people of interest. There's no better way for a team or organisation to get an easy overview of what to do, whats in progress and what someone might need help with.
Even the completely free version has enough functionality to work for smaller organisations. This makes it easier to get started and to, in later stages, motivate using a paid plan. If you have a good enough service, you don't have to limit the functionality very much - customers will stay and pay regardless.
I also use this personally, outside of work, to keep track of my schedule and to plan events with friends and family. I've been using Trello regularly for a couple of years now and my hope is to keep doing it - and even implement it at my current workplace, if possible.
There are tons of templates for almost any occasion, which makes it easy to get started. You pick something that works, and adapt it after your own needs.
I have, however, not used their customer support and cant say weather it's good or not.