Feed it anything at all — big, small, negative — and it hands back a number between 0 and 1.
The S-shaped curve is the sigmoid. Your input x enters along the bottom, rises to meet the curve, and leaves along the left as the output σ(x) — always somewhere strictly between 0 and 1.
Drag x and watch the output. Press Send a number to follow one value through the whole path.
Around x = 0 the curve is steep: a small change in the input moves the output a lot, and 0 maps exactly to 0.5. Drag x far out either way and the curve flattens — the output crowds up against 1 or down against 0 and barely moves at all.
That flattening is called saturation, and the second bar measures its cost. The steepness is the learning signal a network gets back, and it peaks at just 0.25. Out in the flat regions it falls to nearly nothing, so a deep stack of sigmoids passes almost no signal back to its early layers — which is why hidden layers usually use something else, while a final yes/no layer still uses this.