A simplified explanation of how seed phrases work is that the wallet software has a list of words taken from a dictionary, with each word assigned to a number. The seed phrase can be converted to a number which is used as the seed integer to a deterministic wallet that generates all the key pairs used in the wallet.

This compares well with writing down a raw private key where a single letter being unreadable or incorrect can make the private key useless (depending on the serialization format).

A deterministic wallet is a system of deriving keys from a single starting point known as a seed. The seed allows a user to easily back up and restore a wallet without needing any other information and can in some cases allow the creation of public addresses without the knowledge of the private key. Seeds are typically serialized into human-readable words in a seed phrase. The BIP 0032 standard for hierarchical deterministic wallets is used by all good wallets as of 2019.

This BIP describes the implementation of a mnemonic code or mnemonic sentence -- a group of easy to remember words -- for the generation of deterministic wallets.
It consists of two parts: generating the mnemonic and converting it into a binary seed. This seed can be later used to generate deterministic wallets using BIP-0032 or similar methods.