Why you should take a rest day
Whilst being active to some degree everyday is important and carries a range of benefits for both your mental and physical health, over exerting yourself (be it in the gym, on the track, on the pitch, in the ring etc.) every single day would not be advised.
More is not always better and, despite what certain social media and other media outlets may promote, "getting after it" on a 24/7 basis without any form of programmed recovery is likely going to end up doing more harm than good.
What is a rest day and why are regular rest days important?
The old rest day has gotten a bad reputation; again, this may be social media driven, but it's certainly not "lazy" of you to take a rest day, nor do you "want it less" than the other person (imaginary or not) you're competing with.
Rest days are planned and purposeful; they're a programmed day of rest that allows you to physically recover from and adapt to your activity / activities of choice as well as allow you to recharge your mental "batteries". Whilst delayed onset muscle soreness (DOMS) is an indicator of muscle damage (an important factor in muscle growth), to continue to push and push when experiencing DOMS may lead to overuse injury.
Rest days are important as this recovery period is actually when the process of muscular recovery and adaptation can effectively take place. During a rest period, your damaged muscle has time to repair and adapt leading to muscle growth and improved strength / fitness level.
Training 24/7 gives your body next to no time to recover and, whilst you may feel like you're getting leaner (which is more a biproduct of a high energy output and likely changes in nutrient portioning and metabolism), you may not necessarily be progressing.
Many of you will have likely seen a decline in your fitness level or performance during strength training when you haven't been able to achieve adequate muscle recovery.
Rest days will prevent muscular fatigue and loss of performance whilst also contributing to a minimized risk of injury. Additionally, they can also contribute to better sleeping patterns as poor sleep quality is associated with overtraining (likely due to changes in immune function, changes in stress and anxiety as well as overall changes in mood state).
Overreaching is a strategy some coaches will use which increases training beyond recovery capacity for a short period of time ultimately leading to superior adaptation (in some cases) following rest.