Gymnasts, martial artists, ballet dancers and some athlets (e.g. javelin throwers) stretch to increase their range of motion because the latter is beneficial for their activities. The effect is temporary (+) unless you start at a young age and ruin your joints (-> China). If you don't belong to those groups, stretching doesn't do much for you, as has been repeatedly pointed out in various publications. The other benefits attributed to stretching, e.g. a lower risk for injuries, are actually caused by the warming-up effect of the stretching movements and can be also achieved, in a much safer way!, by some simple exercises.

(Side note: When talking about the benefits of stretching, some people also mention Yoga, but that's a different story and 99.999% of Yoga teachers and practitioner don't have the slightest idea what they are doing.)

(+) That's the reason why you have the impression that you are stiff when you skip stretching for one or two days.