Kennings, or circumlocutions, are fun indeed! You're quite correct about their function: first of all they're a way for the poet to flex his or her poetic muscle, show off their knowledge in mythology and to give the reader a moment of understanding, especially when it comes to the complicated ones. Without a good knowledge on Nordic mythology there's no way of understanding a large amount of them, and even with the ones that are more straight-forward you'll have to brain a bit to figure out the logic behind them. Not to mention some were era-specific in ways that nowadays make originally easy kennings almost impossible to understand without explanations.
It's thanks to kennings that we have Snorri's Edda by the way. Himself being a Christian his motive was not to just jot down the old beliefs for fun, but to list enough information for future poets to come that they'd be able to keep up the art that at Snorri's time was already fast on its way of dying out. Iceland was almost the only place left where people still knew how to create certain types of poetry, which made Icelanders quite popular in courts outside of Iceland. A young man capable of konungsdrápa, a type of poem meant for praising a certain king, could easily find themselves employed by a king abroad.
Mythology based kenningsThese are the very kennings Snorri wanted to preserve. The aforementioned Þrymskviða would f.ex. have a person called Jarðar bur, son of Earth. To understand who that is the reader/listener has to know that Jörð is a goddess who had the god Þór with Óðinn, thus Jarðar bur = Þór/Thor. This is a very common type of a kenning by the way, referring to a character via their parent; Laufeyjar sonur (= Laufey's son) also gets a mention in Þrymskviða and it doesn't take long to figure out that that's Loki. Mythology-based kennings can also include pointing to better-known actions from mythology, such as Baldur's bane = mistletoe; the only thing that could harm Baldur was mistletoe and in the end he was killed by an arrow made of it.
Kennings could be made out of lesser-known parts as well, as long as the listeners could make a connection between the kenning and its meaning. One way was to use something that was already in the poem itself - alsnotra ambátt (= wise slavegirl) from Þrymskviða is also Loki, who basically jumped at the chance of dressing up in drag just to follow Thor to Þrym's place. This kenning would not be understandable outside of this particular poem though, unless you first referred to Þrymskviða enough for the listener to figure out which slavegirl that would be.
Simple kenningsI chose to call these simple, but by that I mean the kenning only includes ONE kenning - there's some that are crazy long! These simple ones can be easy-ish to understand, such as wave's horse = ship, or spears' crash = battle. Goddess of rings is another common one and simply means a lady, typically the composer's own wife. Then there are harder ones: war leek/onion (= sword) is my own personal favourite.
Or war-gull, which means raven. Actually any bird coupled with the word 'war' means a raven since they were a common sight on battlefields for obvious reasons... likewise raven's food = warrior, and raven's wine = blood. Simple kennings can be strung together, a technique used especially if the first one can mean several things: "the possessor of the fallen slain and the owner of Sessrúmnir" is actually two kennings after each other. The first part is not clear enough on its own, since Óðinn and Freyja both possess fallen warriors. The second one pinpoints the person as Freyja, Sessrúmnir is her hall after all.
Double/triple/so forth kenningsThis is a different type than just several simple kennings strung after each other in that the extra ones are built into one big circumlocution. There's no limit to how many kennings one circumlocution can include, and it's these ones that really challenge the audience. Hawk's land's fire = golden arm ring, but how? To begin opening the kenning you'd first have to know that hunting with hawks was a popular sport for the nobles in Medieval times, and since hunting hawks perch on your arm a hawk's land/cliff/mountain means an arm. Fire can mean gold when used in a kenning, and it works here because the listener has already made the connection that hawk's land means an arm, so what's the fire attached to it? Jewellery, naturally. Some are taken to be so obvious that they can be used to mean something quite different than what they sound like: "destroyer of hawk's land's fire" = "destroyer of arm rings" = a very generous person... because giving away your arm rings was a very generous act.
Modern useKennings never went away in Iceland, and people understand many without even having to figure out the basics. "Óðs manns æði" means total insanity, or literally translated Óð's man's frenzy. Óð here is Óðinn himself and the madness described stems from the fact that Óðinn worshippers were considered the craziest on the battlefield, looking to die in a spectacular manner in hopes of convincing Óðinn to take them to Valhalla. Likewise the song "Hættu að gráta hringaná" (= don't cry, hringaná) that was written to comfort a poor servant girl who had to have her toes amputated because of frostbite and was crying over the loss. There's a hidden kenning in hringaná: it's shortened from "hringa Gná" (= Gná of rings), in which form it suddenly becomes a very typical kenning that means simply "a lady". The composer was even clever enough to use the name Gná, a goddess who serves Frigga by running her errands, to make the connection to the servant girl obvious.
So in short kennings are a kind of a game between the poet and his/her audience where poet makes a riddle and the listeners try to figure it out, only they have very, very little time for guessing so the kennings have to be just the right level of difficulty. Make too simple ones and no one's impressed, make too difficult ones and no one gets what you mean. But to create one that opens up to your listeners in a moment of "HAH, WOW!" is when you've truly succeeded in making one.
EDITing to add: I have no idea how I managed to ignore pointing this out, but kennings are often vital for creating poetry that fits certain traditional Icelandic metres that rely on alliteration, such as f.ex.
rímur.
