Factor analysis is a statistical method for finding a small number of underlying dimensions that explain the pattern of correlations among a larger set of measured variables.
Here's the intuition: if you ask people hundreds of questions about their personality — do they enjoy parties, do they speak up in meetings, do they seek out new experiences — many of those answers will move together. People who enjoy parties also tend to speak up more. Factor analysis looks at which questions tend to rise and fall together across thousands of people and groups them into a smaller number of underlying factors.
This is exactly how the Big Five was discovered. Researchers didn't start with five traits and go looking for them — they started with hundreds of personality-describing words and traits, ran factor analysis on how people rated themselves and others, and found that the correlations consistently collapsed into five broad clusters.
Factor analysis doesn't tell you what a factor means — that's a human judgment call made after the fact, based on which specific items load most strongly onto it. Two researchers can run the same analysis and reasonably disagree about what to name the resulting factor.