Cross-training provides two potential benefits, one mental and one physical. It can potentially reduce the risk of injury and, as others have already mentioned, it can provide some mental benefit by reducing boredom and burnout through different physical activities.
As this article notes:
Cross training is a great way to condition different muscle groups,
develop a new set of skills, and reduce boredom that creeps in after
months of the same exercise routines. Cross training also allows you
the ability to vary the stress placed on specific muscles or even your
cardiovascular system. After months of the same movements your body
becomes extremely efficient performing those movements, and while that
is great for competition, it limits the amount of overall fitness you
possess and reduces the actual conditioning you get while training;
rather than continuing to improve, you simply maintain a certain level
of fitness. Cross training is also necessary to reduce the risk of
injury from repetitive strain or overuse.
The article goes on to note a few different areas in which to cross-train:
- Cardiovascular exercise
- Strength training
- Flexibility (stretching, yoga)
- Speed, agility, and balance drills
- Circuit training, sprinting, plyometrics and other forms of skill conditioning
Runner's World also had an article discussing the benefits of cross-training from a runner's perspective that may apply to other sports as well:
- Injury Prevention
- Rehabilitation
- Greater Running Fitness
- Active Recovery
- Enhanced Motivation
- Rejuvenation
- Enjoying Other Sports
- Fit Pregnancy
Finally, there are anecdotal stories about the increase in young athletes getting injured at higher rates due to sports specialization. One article discusses some of the dangers of specializing in sports at an early age, while another article quotes Pittsburgh Steelers safety Troy Polamalu:
"I didn't just play football as a kid," he told ThePostGame.com
recently. "I played soccer, baseball, basketball as well. It made me a
far better all-round athlete than I would have been otherwise and it
is the reason I am in the league. I don't like to see kids playing
just one sport. It is like they are pros from the age of 9 or 10. I
don't like it and I don't understand it."
In short, cross-training can provide mental and physical benefits by improving mental health and physical fitness while reducing the risk of injury from overuse. Specializing strictly in one sport may bring about a faster increase in performance, but the articles above suggest that the performance and skill improvements will plateau and may lead to greater injury risks.