The miner "collects" transactions just by listening on the peer-to-peer gossip network where nodes send transactions to one another.
Every transaction has a digital signature which proves that this specific transaction was authorized by the owner of the funds (specifically, the person who knows the private key corresponding to the address where the funds were previously held). You can't change the amount, destination, or fee of the transaction without invalidating the signature. So the miner doesn't have to trust that all received transactions are valid; she can (and must) actually verify the signatures.
There's no guarantee that all transactions sent over the gossip network are "authentic", but as noted above any inauthentic transactions would be immediately detected and ignored, due to the signature being invalid. Moreover, nodes that send such transactions get banned by their peers - this isn't strictly necessary for security (since everybody is verifying signatures anyway) but just helps keep those nodes from wasting everyone's time and bandwidth.
Likewise, when a miner produces a block, other miners (and every other node on the network) will check that the block meets all requirements of the Bitcoin protocol. One of these requirements is that every transaction in the block has valid signatures. So if our miner tries to "distort" transactions, their signatures will be invalid, and so her block will be considered invalid by other miners. This means that when they produce their own blocks, nobody will list her block as the "previous", meaning that it won't end up as part of the block chain in the long run. This in turn means that she won't get the 12.5 BTC block reward, nor the transaction fees associated with her block. So trying to mess with transactions in the block won't work, and carries a severe penalty besides.