An hypothesis states a theory in a way that makes it testable. In order for an hypothesis to be testable, we need variables that are measurable. Variables are measures or indicators of the concepts. Variables are called variables because they vary across the people we study. To operationalize a concept is to make it measurable by converting it into a variable.
Some concepts are easy to operationalize, such as age, earnings, years of formal schooling, etc.
Others might be more controversial such as race. We might operationalize self-esteem by measuring it according to responses people give to survey questions about how they perceive themselves. Once we operationalize concepts into variables, we can collect data on them and see how they vary across the population we are studying.