I've been using Firebug for a number of years now, and want to share with you, why I think it's so useful - and far more user friendly than other similar tools, such as the inspect element tool for Google Chrome.
Firstly, if you're new to web design, using Firebug can help you pick apart websites you like, and start to make sense of how they are put together. If you see something that looks good on another website, hover over it with Firebug and it'll show you both the html and css used to create the element.
When your building a site, you'll inevitably run in to the odd problem here and there. Let's say you have an issue with you navbar - perhaps it's not quite sitting in the right place. Using Firebug, you can hover over the element and quickly see how the navbar has been marked up with html and styled with css - far quicker than looking through your code. This allows you to see the problem and fix it.
If working with Wordpress or a CSS framework, there may be areas of the structure you haven't hand-coded. Firebug in invaluable for these situations. Let's say you're using the CSS Framework Bootstrap and you want to change the default colour of the navbar. Again, like the previous example, simply hover over the element, and you'll be able to see exactly what to target in your css and customise the style.
These are just a quick few thoughts. Again I love Firebug and wouldn't be without it. I also find it extremely useful when teaching students, as rather than having to plough through all their code, and can save time using Firebug and error check much more efficiently.