In 1960, the Chiefs were one of the original teams of the upstart American Football League under the ownership of Lamar Hunt, who also founded the AFL. Hunt originally wanted to become an NFL owner in the late 1950's, trying to buy the Chicago Cardinals or create an expansion team.
In the meeting with the Bidwells, the Cardinals' owners, he was told many people wanted into the NFL, thus giving the Bidwell's no reason to sell to Hunt. This made Hunt realize he could organize the potential buyers and create his own league, thus the birth of the AFL.
The NFL tried to sink Hunt by reversing their stance on expansion which was thought of negatively in the late 50's due to booming TV ratings and an influx of cash the old owners weren't ready to share with newcomers. Once the NFL found out Hunt's team was based in Dallas as the Texans, the Dallas Cowboys were born as an attempt to steal attention.
During the three years the Texans and Cowboys co-existed from 1960-62, Hank Stram's bunch were much more successful, even winning the AFL Championship Game over the Houston Oilers in 1962. It would be their final game before relocating to become the Kansas City Chiefs.
In spite of popular thought, the Texans were hanging in quite well with the Cowboys attendance-wise while they shared Dallas. One of the reasons was Hunt's ability to promote. One example would be the Texans telling fans that if they presented a ticket stub from a Friday night high school game, they would receive free admission.
The reason? Many Cowboys games were played on Friday's during their first few seasons. A ticket stub from a high school game meant those patrons weren't supporting the Cowboys.
However, Hunt moved after three years to help the AFL succeed. While he knew the team might succeed in Dallas, Hunt realized it was better for his league if Hunt moved on instead of fighting against the more powerful NFL.
So, when you see the Cowboys come running out on the field at Arrowhead Stadium, thank them. Without Dallas becoming an NFL city, Kansas City wouldn't have the Chiefs to go crazy over every autumn Sunday.
Other notes: All-time, Kansas City is 3-6 against Dallas with a 2-2 mark at home. The last time the Chiefs won versus the Cowboys was back in 1998. Since then, Kansas City has fallen twice, including the famous Miles Austin coming-out party, when he had 250 receiving yards.
Current players who have been on both teams are Brandon Carr, Kyle Orton and Brian Waters, all suiting up for the Cowboys nowadays. Full list of Cowboys-turned-Chiefs and vice versa here.