Bitcoin transactions, or more precisely transaction outputs, are little scripts that are executed in a VM. To spend a transaction output, you have to "solve it"

Bitcoin transactions, or more precisely transaction outputs, are little scripts that are executed in a VM. To spend a transaction output, you have to "solve it" by providing it an input which makes it return true. The most common transaction script checks that you possess a private key through a signature check, but it's possible to make more complex scripts like the "Pay To Multisig" script. Of course, Bitcoin scripts are quite limited and, unlike Ethereum smart contracts, they are non-Turing-complete and can't store state. Permissionless just means anyone can create transactions because there's essentially no way to block someone from doing so, unlike say a transaction on PayPal.