I see mention of this often on this site. Why and where is it done?

marine biology

Cool beans. I wonder if they can track the age of fish that way to some extent?

they can track age and many other variables

Yup. They can do it with the scales (actual fish scales, not the weight measurer).

Tagging fish can be done for a number of reasons but is probably most often used as a tool that can be used to measure/monitor the health of the fishery. On initial tagging, the fish is measured for length, it might be weighed, it's tagged, and then released. On subsequent re-captures of the tagged fish the same data points are recorded and compared to previous measurements, thus providing information about the growth rate of the fish, and in turn providing a measure of the health of the fishery. For water that is actively managed, this data might lead to fishery alterations/improvements like addition of habitat/structure, aquatic vegetation alterations, addition of forage fish, improvements in cover to protect breeding forage fish, etc., etc.

A coincidence that this thread was started as just yesterday I tagged my first fish. The "aquatic consultant" that manages my local lakes asked me to volunteer to tag and recapture fish over the next several years. I have an initial quota to tag 250 LMB in each of two lakes, and tag 500 LMB in a third lake. Dang, drafted to go out and catch, and hopefully periodically re-catch, 1000 bass. It's tough work, but someone has to do it...

My first tagged bass:

I tagged and released a 26" baby cobia from a pier in the Outer Banks last summer. Now if someone else catches it I will get a letter in the mail about the details! Pretty cool.