The present perfect continuous tense (also known as the present perfect progressive tense) shows that something started in the past and is continuing at the present time. The present perfect continuous is formed using the construction has/have been + the present participle (root + -ing).
In this sentence, using the present perfect continuous verb tense 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 in this case, given the length of Tolstoy’s weighty tome).
Recently and lately are words that we often find with verbs in the present perfect continuous tense.
Of course, not all verbs are compatible with continuous action. Some examples of such verbs are to be, to arrive, and to own.