えぐり is the 連用形 (continuative form) of えぐる, which means "to hollow out", "to gouge"
えぐり + 込む is used asa 複合動詞 (compound verb), which can be attached to the continuative form of another verb. Such verbs, used asThe second verb (後項動詞) in a compound verbs, lose theirusually loses its original meaning. A small list of such verbs can be found in this fluentu article.
込む alone can be translated as "to be crowded", whereas this meaning is lost here and it's more something like "to do something completely", "to do something extremely". A great website for understanding those verbs is http://vvlexicon.ninjal.ac.jp/db/, where you can click on "Search by V2" and search for 込む. There you will get plenty of different examples with verb compounds using 込む, covering all the different nuances that 込む might have.
べし is the same as べき, which means "to be supposed to do", "to should do something" in this case.
~ようにして打つべし becomes "you are supposed to hit as if you were hollowing out something completely"