In short, it is possible to make valid copy of a transaction by modifying a signature in a certain way, and the copy has a different transaction ID. Now, that is not a problem for Bitcoin because only one of the copies will make it into the block chain and the others will be forgotten. The problem occurs in software that monitors transactions using the transaction ID.
For example, suppose a customer of an exchange wants to withdraw some bitcoins. The exchange creates a transaction and records the transaction ID. Normally, the exchange debits the customer's account when it sees the transaction ID in a block. However, if a copy of the transaction with a different transaction ID is included in the block instead, then the exchange doesn't know that the bitcoins were received and it won't debit the account as it should.