For new transactions to be confirmed, they need to be included in a block along with a mathematical proof of work. Such proofs are very hard to generate because there is no way to create them other than by trying billions of calculations per second. This requires miners to perform these calculations before their blocks are accepted by the network and before they are rewarded. As more people start to mine, the difficulty of finding valid blocks is automatically increased by the network to ensure that the average time to find a block remains equal to 10 minutes. As a result, mining is a very competitive business where no individual miner can control what is included in the block chain.
The proof of work is also designed to depend on the previous block to force a chronological order in the block chain. This makes it exponentially difficult to reverse previous transactions because this requires the recalculation of the proofs of work of all the subsequent blocks. When two blocks are found at the same time, miners work on the first block they receive and switch to the longest chain of blocks as soon as the next block is found. This allows mining to secure and maintain a global consensus based on processing power.
Bitcoin miners are neither able to cheat by increasing their own reward nor process fraudulent transactions that could corrupt the Bitcoin network because all Bitcoin nodes would reject any block that contains invalid data as per the rules of the Bitcoin protocol. Consequently, the network remains secure even if not all Bitcoin miners can be trusted.