> How can the top statement be equal to the bottom?
It isn't. Consider ¬A ∧ C:
A ∧ (¬B ∨ C) = ⊥ ∧ (¬B ∨ ⊤) = ⊥ ∧ ⊤ = ⊥
C ∨ (¬B ∧ A) = ⊤ ∨ (¬B ∧ ⊥) = ⊤ ∨ ⊥ = ⊤
Also: truth table:
A B C | A∧(¬B∨C) C∨(¬B∧A)
------+------------------
F F F | F F
F F T | F T *
F T F | F F
F T T | F T *
T F F | T T
T F T | T T
T T F | F F
T T T | T T