Recursive functions are an interesting idea in programming. They allow a function to call itself to solve problems. Here are a few easy-to-understand examples of how recursion works:

1. Factorial Calculation
   One classic example is calculating the factorial of a number. The factorial of a number $n$ (written as $n!$) is found by multiplying $n$ by the factorial of $n-1$. The simplest case is when $n$ equals 0, where $0!$ equals 1. This shows how each function call helps in reaching the final answer.

2. Fibonacci Sequence
   The Fibonacci sequence is another great example. In this sequence, each number is the sum of the two numbers before it, usually starting with 0 and 1. We can find the $n$-th Fibonacci number using recursion with this simple formula:
   $F(n) = F(n-1) + F(n-2)$
   Here, $F(0) = 0$ and $F(1) = 1$ are the starting points.

3. Directory Traversal
   In computers, recursive functions can make it easier to look through files in a folder. With recursion, a function can list all the files in a folder, and then call itself for any folders inside, making the code shorter and easier to use for any nested structures.

4. Game AI
   Recursion is also useful in game AI. In many games, AI uses decision trees to figure out the best move. Functions can check different moves and their results by calling themselves, helping the AI decide the best strategy based on what's happening.

These examples show how recursion can make solving problems simpler and clearer. Learning how to use recursion is an important skill for anyone studying programming!