In the Ragnarok events (Marvel Comics), Bifrost Bridge is shattered. How did this happen?
Also, providing Bifrost isn't used to travel across realms in comics, what's the significance of this?
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In the comic universe, the Bifrost is what anchors Asgard and Midgard together. It's the primary means of travelling between those two worlds. There are other portals in Asgard to various other realms, including places like Olympus that are not part of the Nine Realms. The Bridge was the only permanent one, and allowed free passage in both directions (assuming you could get past Heimdall.)
It should be noted that, in the comics, it's explicit that Asgard is a different dimensional plane than Midgard; some of the Nine Realms are on the same plane as Asgard, and thus can be reached via normal travel. Others, like Midgard and Jontunheim, are on their own planes, and need portals to reach them. In the MCU, there are strong indications that all Nine Realms exist in the same dimension, just different regions of the universe.
In keeping with the mythological meaning of Ragnarok, meaning the end of the world, the shattering of the Bifrost was part of the total desctruction of the Asgardian realm, sending all of the Asgardians into eternal sleep. Some, like Thor, are eventually reawakened, but the believe that Midgard is the only one of the Nine Realms left. I suspect that's not literally true, since Ragnarok is a cycle, not an event, though it's been claimed that Thor "broke" the cycle this last time. At the very least, though, Asgard as we knew it is unreachable from the Earth side.