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Remember what the backwards signal is actually asking at each step: "if I nudged this neuron a little, how much would the answer change?" That is the whole job of the training signal. It measures how much a small change here matters, so the network knows how hard to adjust each weight.
Now look at one of those flat ends of the S. Where the curve is nearly flat, nudging the input barely moves the output. A big push in, a tiny wiggle out. So the answer to "how much would a nudge matter?" comes back as "almost nothing." And a sigmoid spends most of its range out on those flat ends.
That "almost nothing" is a small fraction, often around 0.2. So every time the backwards signal crosses a sigmoid, it gets multiplied by roughly 0.2. Multiply by a fraction and a number shrinks. Then it reaches the next layer and shrinks again. And the next.
Send the correction back through the network and watch where it goes. It leaves the output at full strength, but keeping only a fifth, then a fifth of that fifth, then a fifth again, by the time it reaches L1 at the front there is almost nothing left. The signal does not fade gently. It falls off a cliff.
That is the vanishing gradient, in plain arithmetic: a bend whose flat ends multiply the signal down, layer after layer after layer, until the front of the network is learning from a whisper of a whisper.