The sports coaches believe that by lifting heavier the athlete will bulk up and the increase in muscle mass will make the athlete inflexible. This is completely wrong , what makes you stiff is the avoidance of flexibility workout or what we call as stretching after the workout.
There is always some amount of tension remains in your muscle which in the medical world we call TONE(thats where the words like tone my arms came). This tension helps the muscle to contract and once the contraction is done the tone decreases.So when you do a weight training session this tone/ tension increases and it leads to a stronger contraction. Once you are done with your session if the tension is not released will lead to stiffness( remember the first time you did bicep curls and you can't straighten your arms). With time the body adapts to the tension so you don't feel stiff right after the workout. This tension slowly builds up leads to muscle stiffness.
If you don't want to feel the stiffness ,do stretch after your workout. It will not decrease your muscle size or muscle pump but will help you to move better.