The present perfect continuous (also known as the present perfect progressive) is a verb tense used to talk about something that started in the past and is continuing at the present time.
In this sentence, using the present perfect continuous conveys that reading War and Peace is an activity that began sometime in the past and is not yet finished in the present (which is understandable, given the length of Leo Tolstoy’s weighty tome).
How to form the present perfect continuous
The formula for the present perfect continuous tense is has/have been + [present participle (root form of verb + -ing)].
Recently and lately are words that we often find with verbs in the present perfect continuous tense.
Not all verbs are compatible with continuous action. Verbs that describe states and conditions, such as to be and to own, for example, do not make sense in the present perfect continuous tense. When you want to show that what is being described by one of these verbs continues up to the present, you use the regular present perfect tense.