In Europe, the Western Roman Empire had crumbled under its own weight. Europe went through a period of chaos as German tribes picked the bones of the Roman Empire clean. Out of this 'dark age' new powers emerged; France, England, the Holy Roman Empire, and Spain to name a few. While these looked like nations when we observe maps depicting these times, they were in fact composed of tangled webs of personal and political loyalties that shifted and changed. As a result families often meant more politically than borders; there was a time when the King of England ruled more of France than the King of France through family connections.
Likewise half a world away, in the islands of Japan, the land was divided between many different kingdoms, connected by a similar web of complex political, economic and personal relationships. The ruling class were the land owning lords, known as Daimyos.
Warfare was a constant in this ever shifting landscape and in both Japan and Europe warrior classes emerged. Daimyos in the East and Feudal Lords in the West gave plots of land and peasants to warriors in exchange for their service. These warriors, free of the need to harvest their own food, were able to devote their time to training for war. These grew into their own warrior cultures with unique practices and ideologies.
In Japan, they were the Samurai and followed the Code of Bushido, while in Europe it was the Knights who had the loose ideology of Chivalry. Both Knight and Samurai used swords, spears, and other weapons to defeat their enemies, and they shared one more quality: they wore armor. This armor was the best protection technology could provide and would stop not just glancing blows, but direct strikes from all but the heaviest weapons.
How to defeat an armored opponent became the question that martial skills sought to solve. Punches and kicks were not very effective against armor, so grappling became one answer. In a clash of swords, one fighter could quickly position himself and throw his enemy to the ground. The enemy, once on his back, is weighed down by his armor and then attacks could be directed at the gaps in the armor with the added force of gravity to help break through.
(a good video of "Wrestling at the sword" in armor)
(an example that really demonstrates the speed and violence grappling could bring to a sword fight)
Both arts developed their grappling in societies where practically everyone was armed, so the focus was defending against attacks from blades. Dealing with attacks from knives was at the core of both Jujitsu and Wrestling and the vast majority of techniques are the same.
Here is a shoulder lock takedown used against a dagger attack in both Jujitsu and European wrestling:
European over hand attack dagger defense.
Uke Garami against over hand knife attack (via nidanwarrior11)
These techniques are not just similar, they are identical in principle and practice. The jujitsu technique is one of the oldest in the martial arts and the European one comes from the combat manual Flos Duellatorum based on German fighting techniques and written before any Europeans ever had contact with the Japanese. Thus completely independently from each other, the Japanese and Europeans came to the same technical solution. And it is not just restricted to this technique, there are many similarities, either identical techniques or ones that work on the same principles.
Japan's warfare gave way to a time of peace and during this time the Samurai began to open schools to train their styles of combat, including Jujitsu. Despite being at peace, Japan was still divided and Jujitsu schools had little contact with each other and thus the styles grew apart. Japan began to have contact with European explorers and traders, but a closed door policy was imposed by leaders, locking out foreign influences.
While Japan looked inward, Europe looked to expand outward. Europeans traveled the world on the ships and those same ships that carried explorers and merchants also carried wrestlers. Wrestlers, who in the endless hours to fill ashore, would often challenge local wrestlers to matches and a lot of spectators would bet on the results. As a result Persian style wrestling from the Middle East and Indian wrestling from the sub-continent began to blend with the European Wrestling.
Back in Europe, wrestling was still very popular. In fact it became common custom for the Kings of European nations to wrestle each other at festivals. In one rather famous encounter King Henry VIII of England grappled with King Francis I of France. They both practiced wrestling arts that used jackets (similar to gis) and wrestled to the first flying throw (similar to an Ippon), so the match would have looked very much like a Judo match. King Francis won by a Flying Mare throw (similar to a Seoi Nage).
But the advent of the gun brought to an end of battles fought primarily with close combat and armor became a thing of the past. In Japan, the Meiji Restorations put an end to the ways of the Samurai and Jujitsu would have faded into the past but for a man named Kano Jigoro. He took the divided and complicated art of Jujitsu and smooth lined it into the modern sport of Judo. Some pockets of Jujitsu survived and are taught to this day in its pure form.
In Europe, it was purely a matter of practicality and not government policy that relegated wrestling to sport instead of combat art. Wrestling's divergent into several different styles of sport wrestling including what would become Olympic free style wrestling and catch wrestling, a grappling art that included submission holds.
While Jujitsu and catch wrestling existed for centuries in their home lands it was not until these two arts switched hemispheres that would lead directly to the birth of the sport of Mixed Martial Arts.
Special Thanks to KJ Gould for his help with catch wrestling history.
For more on the rise of Jujitsu and Judo in Japan: