how are real numbers used to describe real-world situations?

In practice to get real-world answers only rational numbers are used. For example the irrational real number π is replaced by a rational approximation such as 3.141593. However irrational real numbers might still be used in descriptions. For example the area of a circle whose radius is 8cm might be described as 64π cm^2.

-To make measurements
-To display monetary value
-To denote scales of performance, satisfaction, symptoms, or just about anything else

Ultimately, real numbers describe the quantity of something and are used for this reason, whether the quantity be of a physical object or an abstract concept.