Two words: blockchain technology.
Transactions in bitcoin are encoded, packed into a block of other transactions, and all of these are sent out to thousands of computers running blockchain computations, known as hashes. All of these computers are running similar algorithms designed to force honest work and to take time for the computers to complete.
The purpose of this step is merely forcing the blockchain to require time, energy, and effort, and to be randomized and decentralized when it is validating transactions. Whichever computer solves it first receives an incentive reward, and the entire blockchain, comprised of all computers running bitcoin client software, then updates the ledger to include the most recent validated transactions.
The ledger is distributed across all of the decentralized nodes and cannot be changed without being detected. On the user side, the transaction is what’s called a push transaction, which means that the funds are sent from the payor’s account without having to be requested from the payee. With bitcoin, the blockchain validates transactions in an average of 10 minutes.
While this is seen as a necessary process by most proponents of bitcoin, many offshoots and other cryptocurrencies are designed to solve blocks in shorter time spans. When the transaction is validated, the balance in the payor wallet drops and the balance in the payee account increases at the exact same time, as far as the distributed ledger is concerned. The system is designed to require no trust between the individual parties or individual computers in the blockchain since the system as a whole maintains trustless security by consensus in the ledger.