One can always fear not to have checked correctly all the lines of the truth table.
So the most secure method is to rephrase the " tautological consequence test" in terms of corresponding conditional.
The assertion " Q is a tautological consequence of premises P1, P2, P3...Pn " is equivalent to the assertion
" the conditional formula [(P1&P2&P3&...&Pn) --> Q ] is a tautology ".
Using this rephrasing , you make a truth-table for this last conditional formula, and you only have to look at the last column of your truth table. If you have only the truth value T in this column, it is guaranteed that Q is a tautological consequence of your set of premises. If you have at least one time the value F in the last column, you know Q is not a tautological consequence.
Now, to answer precisely your question ( regarding what you point as difficult to understand). Knowing that ( A --> B) is the same thing as "it is not the case that A is true and B is false", you can seee that the following 4 assertions are equivalent.
(1) the formula ( P1&P2&P3...&Pn) --> Q ) is a tautology,in other words, is true in all possible cases ( all possible " interpretation")
(2) For all possible interpretation i if premises P1, P2, P3...Pn are all true in i , then Q is true in i
( Here, we quantify over the set of all possible interpretations)
(3) There is no possible interpretation i such that P1, P2, P3...Pn are all true in i and Q is not true in i
( In order to pass from (2) to (3) we use the predicate logic equivalence : For all x, phi(x) <--> there is no x such that ~ phi(x) )
(4) The set of interpretations in which premisses P1, P2, P3...Pn are all true is included in the set of interpretations in whic Q is true.
( Here, the universe is I= the set of all possible interpretations)