I have just looked at 20 or so sentences containing this expression (Google search). All the sentences are in formal/ academic style. In each sentence, this expression can be replaced by "how much". In everyday English we would say "how much" - e.g. "I don't know how much the guests will enjoy my my new recipe. The combination of tastes is a bit unusual." More formally: "We are uncertain of the extent to which the guests will appreciate the new combination of unfamiliar tastes".

Hello Dimin, as you probably already know the extent of smth is the limit to which, the range, or ultimate reach. So in layman's terms it means, the scope (how frequent or not). For instance, the sentence "We have no clear overview of the extent to which health-care providers involve patients in the decision-making process during consultations." means we have no idea how frequently (i.e once a week, sometimes, never etc.) health care providers involve their patients in the decision-making...