A full node is a computer running full node software like bitcoin core that has verified every transaction in the blockchain and continues to verify new ones as and when it hears about them. Every full node validates transactions and rejects double spends. If it has learned of a transaction that spends certain coins it will reject any new transaction that tries to spend those same coins unless the original transaction has been marked as replaceable.
Because of the nature of computer networks it is possible that not every node has heard of every transaction. It is also possible that different nodes will have heard of the same transactions in different order. The blockchain fixes the order and history of the transactions so that everyone agrees on the state of the ledger. Once a transaction has been committed to the blockchain and enough blocks have been added to the chain after the block containing the transaction that transaction cannot be modified.
So every node individually prevents double spends that it hears about and the collective consensus of what transactions happened or did not happen comes through the blockchain.
What exactly the process that minors follow to detect that a certain transaction is spending the bitcoins twice?
Miners aren't the only ones validating transactions. Like I said every full node validates transactions to make sure that there are no double spends.
Miners follow the same process as full nodes. The full list of validation rules is given here.
Why don't minors (voters) simply accept a new block without even check if it's truly valid or not? what are the consequences?
There is no voting in bitcoin. It's not just miners who need to check the validity of blocks. Full nodes run by users check it too. If a miner generates an invalid block then full nodes will reject it and the miner will lose money because he will have wasted electricity and time producing that block.