Trial and error. I've done 3D modeling before so the concept of nodes and what have you are pretty straight forward to me - vectors are kind of like the 2D equivalent to 3D modeling. You start with nodes (points) and connect them with segments (lines) to make shapes. Then you fill the space in between with colour data.
Vectors aren't like other images, which use raw pixel data which scales horribly. They use precise scaleable data that can be scaled up/down however much you want without losing any quality whatsoever. This is why Big Ant favours this method because they could in theory scale the game up to 4K or beyond and all the logos would look just as great as the day they were made.
As for using Inkscape. It works pretty much like photoshop would. I can't really say more than that. I just set a jpeg or PNG as a background layer, then use the line tools to trace the logos. Plenty of great guides out there for Inkscape too.