We do not know.What we know, it happened during the collapse.Some say the darkness struck the traveler, ripping apart that shard.What other people say, and what I think would be really cool, is related to rasputin.The theory goes with the darkness about to strike earth. Rasputin, worried about humanity, thought the traveler would leave us, and its no big surprise it did, it already had left the fallen to die.In an attempt to anchor the traveler,Rasputin launched missiles at it, but not to kill it, just to injure it so it wouldn’t leave.The missiles struck, ripping a puece out of it.The traveler, with no option left, had to fight the darkness in orde to protect itself, and also protecting humanity along the way.

Assuming the traveller did the explode thing in the collapse like to kill Ghaul, which cracked it open. But only broke the bottom of it off.

Well, just part of the way they built the universe honestly. If it landed in Russia instead and it was the RDZ, you'd ask the same locations question. It's just how it happened, and it makes sense when the massive Traveller rested over Russia, that pieces would fall in Europe, Asia, Africa, etc. As for corruption, the Taken, Hive, and Fallen have been trying to use the remnant powers in it for a while if you listen to dialogue. It was bound to be tainted somehow.

It probably just broke off and fell there.

There are several reasons why it could have happened. The Traveler sustained damage by the Darkness during the collapse. It's possible that the shard became corrupted by something the Darkness did. The Traveler damaged itself when it pushed the Darkness away. As Tyra Karn says, the shard has been away from the Traveler for too long and its purpose has become corrupted over time. Rasputin attacked the Traveler after it defeated the Darkness, when it was weakened after expending much of its Light. Over time the shard became corrupted.