What happened to permit Thorin to fit the key into the key hole? What rune and moon letters prophesied this?

The key to which you refer is a small, “long-barrelled” silver key that Gandalf gives to Thorin at the meeting in Bilbo’s home, along with the map to the secret entrance to the Lonely Mountain. Thorin closely guards the key by hanging it on a chain around his neck throughout the entire quest.
When the group of dwarves arrives at the Western side of the Lonely Mountain, the location of the side-door according to the map, they find the door but cannot open it. The determined dwarves are “too eager to trouble about the runes or the moon-letters” and attempt to force the door open without success.
After a dispiriting time camped out at the secret door, Bilbo observes a thrush tapping at the smooth surface of the entrance. Suddenly, he remembers the translation of the moon-letters.

Stand by the grey stone when the thrush knocks, and the setting sun with the last light of Durin's Day will shine upon the key-hole.

As Bilbo rushes excitedly to the dwarves to remind them of the prophetic runes,

There was a loud crack. A flake of rock split from the wall and fell. A hole appeared suddenly about three feet from the ground.

Thorin uses the near-forgotten silver key to finally open the door and the party of treasure-hunters is back in business. I hope this helps!