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Code signing attaches a certificate and a signature to software so your computer can check it before installing. The publisher signs the code with its own private key; the certificate authority signs the publisher's certificate, not the code. (The video says the certificate authority signs the code; it does not.) The check rests on public key infrastructure: the operating system ships with root certificates of certificate authorities it trusts, and the publisher's certificate must chain back to one of them, directly or through an intermediate CA. The signature also covers the file's fingerprint, so a file changed after signing no longer matches.
Certificates carry an expiry date. On its own, that would make signed code untrusted the day the certificate expires. A timestamp authority fixes this: it stamps the signature with the time it was made, so code signed while the certificate was valid stays trusted afterwards. Code that was never timestamped loses its trust at expiry.
One limit, straight from the lesson: a certificate does not make code legitimate. Anyone can buy one, so a signature that checks out tells you who signed the code and that it is unchanged, not that it is safe.
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