You need to pick it up. After you purchase it, go to Position -> Items, and you should be able to equip the item from there (or any other items lying around which buyers left behind and did not equip).
Trunks and staves can both be picked up, but you need both hands free for two-handed items, so if you bought too much you may need a place to stow it. If you join a faction and get a ship assigned, you should be able to stash your stuff on board your ship.
Items are not useful yet since combat is not yet implemented, so you can pretty much just carry them around for now for RP purposes, or just to show off if people scan you. :P