Nearly every time I'm mining and a rat comes along it drops a metal scrap in the wreckage when I open the cargo bay. What exactly am I supposed to do with this?
You can reprocess it into Tritanium at a ridiculously useless ratio, but that's about it.
It's pretty much the equivalent of "useless junk" in other MMOs.
If you plan to move to null-sec, save them and bring them with you. One of the problems with mining in null-sec is the lack of low-end minerals (tritanium and pyerite), and these have to be brought to null-sec. Take them to Jita (or your local market hub) and sell them for about 3k each. A single Badger full of metal scraps can haul as much tritanium as 3 Charons.
Moving minerals to null-sec is usually done via 425mm guns - they have the "best" mineral compression that player-made items can do: 28 to 1. Metal scraps (and Reinforced Metal Scraps) have much higher compression: 500 to 1 for Metal Scraps and 2500 to 1 for Reinforced. Reinforced Metal Scraps can only be sold via contracts and can usually net you about 90k per unit.
While it may seem like a lot of work to you - to loot them and haul them to market - they make life much easier for the industrialists in null-sec.