The picture
Start with one large square
A square whose side is a + b has area (a + b)². Instead of multiplying immediately, split both sides at the same point: one part has length a and the other has length b.
Those two cuts divide the square into four smaller regions. Their areas are a², ab, ab, and b².
The identity
The pieces must equal the whole
Putting the four regions together gives (a + b)² = a² + ab + ab + b² = a² + 2ab + b².
The middle term is 2ab because there are two different rectangles with side lengths a and b. Forgetting one rectangle is the most common mistake.
Try a number
Let a = 5 and b = 3
The whole square has side 8, so its area is 64. The pieces give 25 + 15 + 15 + 9, which is also 64.
Use the sliders to try your own lengths. The equality survives every change because the pieces always tile the same square.