How does finding the gradient of a line and the area under a graph relate to real world problems?

Practical applications of calculus occur more often than you would think. The most common example of them being applied to real world problems is through the relationship between distance and time. The distance an object will travel in a certain time is it's velocity. If we differentiate an equation for distance travelled with respects to time, it will give an equation for velocity.
However this is only useful if the objects velocity is constant; it could be speeding up (acccelerating). The acceleration of a body is it's change in speed over time. So, if we differentiate an equation for velocity with respects to time, it will give an equation for acceleration.
You can differentiate again to give the rate of change of acceleration, but this is a less useful figure to be given.
Conversely, if you have an equation for an object's acceleration, you can integrate it (the opposite of differentiating it) with respects to time to give an equation for it's velocity at a certain time.
Integrating again will give the distance it has travelled after a certain time.
Calculus can be used in this way in loads more equations including  force-energy-power, and circumference-area in a circle.
All of these values can be plotted on graphs, and different features of a particular graph (such as the gradient of the line and the area under it) will give you these values. Remember, a graph is just a visual representation of an equation, so that's where they come into it.