Imagine a particle tied to the end of a string going around a circular path on a smooth horizontal surface with a uniform speed. You can work out the speed of the particle by knowing the time taken for one revolution and using speed is distance/ time taken. No big deal.
If you imagine what would happen if the string breaks you would come up with an answer- The partcle travels along a straight line with a constant velocity. Since the direction of velocity did not change after the string broke, the velocity for the straight line motion is same as the velocity at the moment the string broke and so is directed along the tangent at the given moment. Again nothing new! The instantaneous velocity of a body moving along any curved path is of course along a tangent and its magnitude is equal to the speed of travel. You knew that.
But how do you get this instantaneous velocity from the definition of the velocity ?
Since velocity is displacement divided by time, you would need to look at two positions at two different instants to evaluate that. That would infact give you average velocity- not the instantaneous velocity, for that you should work a little harder.
Suppose we did this- consider two instants so very close to each other, that we could not almost not tell them apart. That is the gap between the instants is real small. If we could get the postitions of the moving particle at these two very close instants, we would have almost got the instantaneous velocity. This would of course be an approximation to the instantaneous velocity, but a very good one. By really looking at two very very close instants we would see the value of velocity calculated from the formula getting closer and closer to some value, which could be taken as very good approximaion to instantaneous velocity.
In the applet we are looking at the positions of a moving particle at two different instants. the red vector is the displacement vector and the blue vector is the average velocity vector. We can reduce the interval between the instants, the diplacement vector gets shorter- and it is there for you to see that the magnitude of average velocity approaches a specific value and the direction approaches a tangent. As the denominator term decreases the numerator term also decreases and we can guess that if the duration is zero the velocity at an instant is along the tangent ( Keep in mind that what seems obvious this way is not really that- how come we worked out the velocity when the duration is zero and the displacment is zero; WE can not do that. We really have not done that. We are only saying that it can be approximated to something very reasonalbly close to what we physically perceive and know to be true.)