More speaking time - If 15 students speak in turn in a 30 minute class, that equals 2 minutes of speaking each- and for shy students it might work out as just one minute of speaking or less. If students work in pairs for the whole class, they could theoretically speak for as much as 15 minutes each, and certainly no one could speak as little as 1 minute.
Speaking skills - There are certain speaking skills that happen much more naturally and more often in pairs than in large groups, such as interrupting each other, using tag questions etc. to invite the other person to speak, and asking people for confirmation and clarification of what they said. In general, if you are going to be speaking English one to one in real life the best practice for it is one to one speaking in class, which usually means pairwork.
Cutting down on embarrassment - In a whole class speaking activity, not only do you have the stress of everyone listening to you but there is also usually silence while you are thinking of what you are going to say. In pairwork, the background noise of other pairs speaking to each other can make you much less embarrassed about speaking out.
Fun - Working in pairs allows the teacher to use more games, and therefore increases the students' motivation and concentration.
Individual attention - Although some students think that being asked to work in pairs is a sign of the teacher being lazy or of avoiding talking to the students, in fact the teacher can sometimes ask the whole class to work in pairs so he or she can talk to or listen to one student at a time without everyone else noticing.
Error correction - If one person is talking in front of the whole class and the teacher writes something in a notebook, everyone will know that the person speaking has made a mistake and will know whose mistake it is in the following error correction stage. People working in pairs means that the teacher can more easily hear people's mistakes and correct them later without embarrassing anyone.
Fluency - According to research, students who speak out in front of the class improve their accuracy but lose fluency, and the opposite is true of speaking in pairs. If you need to improve your fluency, this is usually best done with pairwork speaking activities.
Exam practice - Some speaking exams, such as the FCE and CAE, include speaking in pairs. The most efficient way to practice exams that are just one to one speaking with the examiner is also usually to do it in pairs. This is usually done with one student pretending to be the examiner.
Classroom dynamics - Speaking to your classmates individually will help you to get to know them better- and so make the atmosphere in class nicer and the communication between you in all classroom activities more natural.