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In short: Stablecoins stay stable because every digital unit is matched by real money - verified, audited, and redeemable at any time.
Each stablecoin is fully backed by real assets, usually cash or short-term government bonds. 1 digital € = 1 real € held safely in reserve. When people buy, the same amount of real money is deposited and stored by the issuer. When people sell, the issuer returns the money and destroys the corresponding stablecoin - keeping the balance at 1:1.

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The mechanism

Because every coin is backed by equal value in reserves, the price stays stable, even if demand moves up or down. To verify this, issuers publish regular audits and reports, proving that reserves match the total number of stablecoins in circulation.