Ok.
This paragraph is saying that a truth table is not the only way to specify a Boolean function; that a Boolean arithmetic expression can also represent the function.
Line 3 by itself does not establish this fact. That's what the next sentence is trying to say; that all 8 of the lines must evaluate to a match to prove that the expression is an equivalent definition.
This is basically the equivalent of saying that a multiplication table can also be represented the expression x*y.
The table would more clearly show the equivalence if it had been two tables:
Table 1a: Truth table definition of f(x, y, z)
x y z || f(x, y, z)
------||-----------
0 0 0 || 0
... ||
1 0 1 || 1
...
Then present a table showing the equivalence of (x OR y) AND NOT z to f(x, y, z)
Table 1b: Boolean expression definition of f(x, y, z)
x y z || f(x, y, z) || (x+y)~z
------||------------||---------
0 0 0 || 0 || 0
... || ||
1 0 1 || 1 || 1
...