Proving NP Completeness of a subset-sum problem - how?
The trick to the reduction is to use numbers to encode statements about the 3CNF formula, crafting those numbers in such a way that you can later make an arithmetic proposition about the numbers that is only true if the original 3CNF formula is satisfiable.  Using the 3SAT -> SUBSET SUM reduction from your lecture notes:

The $y$ values encode statements about which variables have positive literals in which clauses.  The $z$ values encode statements about which variables have negative literals in which clauses.

The $s$ value is crafted the same way for each clause; put a one in the digit position corresponding to that clause, and zeroes everywhere else.  The $t$ value will be the same as the $s$ value for each clause.

The $k$ value is always 1111... followed by 33333.... where the number of ones is the same as the number of distinct variables in the formula and the number of threes is the number of clauses in the formula.  Note that the required sum $k$ has a one in each digit position corresponding to the variables.  This means that any solution to the subset sum problem can include only encoded statements about either a positive instance of the variable or a negative instance in each clause, not both.  Note also that sum $k$ has a three in the digit position corresponding to each clause.  The $s$ and $t$ values for each clause will sum to two, but to complete the sum a third one will have to come from one of the $y$ or $z$ values.  All three ones could come from the $y$ and $z$ values, but the fact that $s$ and $t$ will only sum to two for any clause guarantees that any empty clause in the 3CNF formula forces the subset sum problem to become unsatisfiable.