We use GTmetrix for two main tasks. It is one of our investigative tools when beginning an SEO audit (or deciding if we even want to take on the client). GTmetrix allows you to quickly get a pretty comprehensive look at the technical health of a website. In just a few minutes you can get a good look to see the data size of a page, get an idea of load time. You can also see if the site is using caching, scaled images, if scripts are minified and/or in-line and the list goes on to address about 25 different parameters.
The other area that we use GTmetrix for is speed optimization. Not only can you see the items mentioned above but you also can look at the Waterfall so you can see the order that assets are loading on the page but also how long each takes. This lets you identify areas that can be improved and find where the biggest improvements can be made with the least effort or cost.
Speaking of cost, all of the above can be accomplished with the free version of GTmetrix.