Area is one simple question: how many unit squares fit inside the shape? A "unit square" is just a square that's 1 by 1 — one centimetre by one centimetre, one metre by one metre, whatever you're measuring in.
Counting, but Cleverly
You could count the squares one at a time, but that's slow and you'll lose your place. Here's the trick: in a rectangle the squares line up in neat rows. If the rectangle is 6 wide, every row has 6 squares. If it's 4 tall, there are 4 rows. So instead of counting all of them, you count one row and multiply by the number of rows: 6 × 4 = 24 squares. Press "show the unit squares" in the toy and you'll see exactly that — rows of squares, and the count matching width times height every time.
So the "formula" Is Really Common Sense
People write it as Area = length × width (or width × height — same thing, rectangles don't mind which side you call which). It looks like a rule to memorise, but you've just seen it's only a quick way to count squares. That's why it never lets you down, and why every other area formula — triangles, parallelograms, all of them — gets built out of this one.
Why "square" Units
Because you're counting squares. If the sides are in centimetres, each little square is one square centimetre, so the answer is in cm². It's not the distance around the shape (that's perimeter, and a different rung) — it's the flat space it covers.