St. Benedict founded the monastery around 529, wrote his Rule here and died here. The abbey has been destroyed and rebuilt several times: sacked by the Lombards in 580 and by Saracen raiders in 884, struck by an earthquake in 1349, and reduced to ruins by Allied bombing in February 1944. The present buildings were reconstructed afterward, and Pope Paul VI consecrated the rebuilt church on October 24, 1964. The Italian encyclopedia Treccani records that the saint's burial niche was among the few parts to escape the destruction of 1944.
There is an old and serious rival claim to the relics. The Abbey of Fleury at Saint-Benoît-sur-Loire in France holds that Benedict's bones were carried there in the seventh century, while Monte Cassino lay abandoned after the Lombard attack. Monte Cassino has always contested this. Reference works differ: the 1911 Catholic Encyclopedia judged the Fleury translation hard to doubt, while the abbey holds the contrary. For a pilgrim the practical answer is simple: you may pray at either place.
Good to know before you go: the abbey stands well above the town, so settle how you will get up the hill. Trenitalia sells a combined train-and-bus journey, called Montecassino Link, by rail to Cassino and by bus up to the abbey; otherwise you need a car, taxi or coach. Come as to a monastery, not a museum: dress as for any church, keep your voice low, and expect parts of the house to be closed because monks live there. The monks celebrate Mass and the Divine Office in the basilica, and the abbey posts its current opening and Mass times on its own website, which is the timetable to trust. The abbey also has a museum.