There is no doubt that the process is not perfect, but it does allow the reader to make some judgment about the relative quality and merit of the research.
Peer reviewing allows a diversity of opinions to be brought to the table, theoretically removing any personal biases and pre-set ideas from the equation.
The peer review process stops a lot of substandard and poor science from reaching publication. In addition, the reviewers are generally experts in their field, well acquainted with the latest developments. They can, therefore, reject duplicate research and plagiarized papers.
Because editors can use the process to remove poor quality work, it saves a lot of wasted time and money, especially if the work is plagiarized. Without referees, a journal would have to employ a team of editors with expertise in every field, and this would make the cost of the production prohibitive.
Traditionally, the journals that use peer review enjoy an excellent reputation and are trusted by experts in the field. This also helps them to attract the best researchers and scientists to submit papers.
The reviewers are experts in their field, and peer reviewing often brings innovative research to their attention, where it may be buried amongst a flurry of papers.
Peer reviewing is not only used for journals but for grant applications and University standard textbooks. This helps to ensure that money is diverted only towards viable research proposals. The peer review of textbooks ensures that students are taught correctly and are provided with excellent information.