In Bitcoin terms, "redeem" means to claim an unspent transaction output (UTXO) by listing it as an input of a new transaction.
All a Bitcoin transaction really does is claim ownership of one or more UTXOs, and combine their value into one or more new UTXOs. To do this, it must prove ownership of the UTXOs being used as inputs. Each UTXO has a script that defines a condition that must be met in order to use it as an input (scriptPubKey or encumbrance script). To redeem, you must supply a corresponding script that meets that condition (scriptSig or redeem script).
The website you refer to is an improvement proposal (that has already been implemented) where instead of specifying a script in a UTXO, you specify a hash of a script. This means that a transaction that claims UTXO needs to specify both the script that meets the condition, and the original script from which the hash was produced.