Since the director left the Witch 2015 horror film open to interpretation, i can only give my interpretation rather than an explanation. I will give my own interpretation behind the film's symbolism, characters, plot, and finally the ending.
The family is banished and as a result, excommunicated from the Church for being falsely accused of witchcraft. I believe that it is the Witch's deception that causes the family to be innocently framed for such an accusation. If you notice in the film, the Witch tricks the family into believing that Thomasin was practicing witchcraft.
The Witch's identity is never fully disclosed but it appears that the Witch is a foreshadowing of Thomasin in the future. This is due to partially to her behavior towards her sister Mercy as she calls herself "the Witch of the wood" and her later transformation.
If she magically stole Samuel who had yet to baptized then it's likely she would've also known about the family's exile and spiritual state. This becomes more evident as she uses her magic to lure Caleb into her home disguised as a seductively beautiful woman.
William, the father of the household in this scene looks similar to that of some painting of Jesus Christ. William often likes to picture himself as the savior of his family and in this sense, he is arrogant. Throughout the film, he often begs for mercy and forgiveness from God but never receives it because he refuses to trust in God's salvation. William wanted to save himself and his family through his own self-righteous piety and it's this attitude that leads his family to starve. At the end, he puts to rest his self righteousness and allows Black Phillip to kill him instead. William finally puts his trust in the grace of God while dying in the process.
What inspired William to take this action? By watching his son die. In Puritan custom, the father was the head of the household as it was in Patriarchal society. Ironically, the son played the role of a spiritual leader for placing his faith in God as signified by his outstretched arms symbolizing a crucified Jesus Christ and dying a peaceful death. It is unfortunate that the family fails to see this and that is what lead to the family's demise except William's at the end.
As it turns out, Black Phillip was Satan in disguised. The black goat in this time period symbolized the Devil evidenced by images such as the Baphomet. It was Black Phillip who was behind every evil act for the sole purpose of luring Thomasin to fulfilling her destiny of a Witch. Why does Thomasin choose this dark path? She could never find acceptance by even her own family. She could never find grace in God's eyes. If she were to go back to civilized society, she would be killed due to being accused of Witchcraft. If she were to live in the woods, she would starve to death. She comes to believe that God has abandoned her. Who does she have left but the Devil?
She goes to a Witches Coven and partakes in the flying ointment in the beginning of the film, completing her transformation as a Witch.
I think this is one of the most disturbing themes of the film but one of the most true. Some will turn to God in the face of tragedy and others will join with Satan. Nevertheless, it carries on a valuable lesson to it's viewers.