Example: When you drive in your car, then your speedometer usually tells you how fast you are going. Speed/velocity is defined as the rate of change of the position. You might measure how far your have driven from a given point. This distance will change over time because you are moving. The rate at which this measurement of position/distance changes is exactly the speed.
Distance/position is then usually (in the US) measured in miles. Speed is measured in miles per hour (mph). That is, speed is what tells you how many miles you are moving per hour. With a constant speed of $60$ miles er hour, in one hour you would have travelled a distance of $60$ miles.
This is just one example of the use of rate of change in every day life. There are many others such as: the rate at which and investment grows, the rate at which the temperature changes, the rate at which the speed is changing (this is called acceleration), etc.
To study these rates of changes you need calculus and derivatives. Why to you need this? Well, because the derive is defined as the rate of change. So b definition studying rate of change is studying derivatives. So, while you might think of derivatives as representing rates of change, they are really defined as such. If you don't want to use calculus and derivatives to talk about rate of change, then what do you want to do?
If you have a function that tells you the position of something (as a function of time), then the derivative will tell you the speed of that something (as a function of time).
Often we are more interested in how fast a quantity is changing that how the quantity is changing.