I'm not much into frontend development, but AFAIU Font Awesome is a bunch of assets packed as a font [0]. Why is this is a good idea? Is it faster, smaller, more convenient, something else?
Font icons are scalable too, which makes it really easy to change the icon size on the fly. Since I do all of my development directly in the browser, I never know how big I need my icons until I've done some experimenting with different sizes. Font icons make this experimentation process a lot easier because I don't need to update an image file. I can just change the font-size property in css.
It's incredibly convenient not to have to generate multiple sizes of icon/pictograms for a project. Beyond even just needing multiple sizes for different uses (a list vs a header) - with the advent of high dpi mobile screens you'll also need to re-generate all of the images in @2x versions if you want the images to not look fuzzy.
Beyond that, there is now a whole ecosystem of addons that make use of the library
One of several different icon pickers: http://victor-valencia.github.io/bootstrap-iconpicker/
Leaflet Map Markers: https://github.com/lvoogdt/Leaflet.awesome-markers
1 - BootstrapCDN will even host it for you: http://www.bootstrapcdn.com/#fontawesome_tab