All of the existing answers talk about magic, but there's also the 'power' of recognition. We are used to an interconnected world with means of authenticating identity using trusted third parties. No such situation in the Tolkienverse.
The brooches (whether or not they possess any particular mystical qualities) along with the cloaks mark the bearers as elf-friends in general and friends of LorienLórien in particular in a world where discerning whether someone you just met is friend or foe might be difficult.
