Could someone please explain to me how (p ∨ q) = (p NAND p) NAND (q NAND q)

For starters, do note that

$$\text{NAND} (p_1, p_2) = \neg (p_1 \land p_2)$$

Hence $\text{NAND} (p, p) = \neg (p \land p) \equiv \neg p$ because $p \land p \equiv p$. Therefore,

$$ \text{NAND} (\text{NAND} (p, p), \text{NAND} (q, q)) = \text{NAND} (\neg p, \neg q) = \neg (\neg p \land \neg q) \equiv \neg\neg p \lor \neg\neg q \equiv p \lor q$$

where I used one of De Morgan's laws and elimination of the double negation (i.e., $\neg \neg p \equiv p$).

Write down the truth table for $p\lor q$.

Then find the truth table for $(p\:\text{NAND}\:p)\:\text{NAND}\:(q\:\text{NAND}\:q)$ and compare. The end.

Here's a proof via the automated theorem prover Prover9: Firstly, we input the assumptions (standard Boolean logic, and define a function as NAND):

formulas(assumptions).

% associativity of "or" and "and"
x v (y v z) = (x v y) v z.
x ^ (y ^ z) = (x ^ y) ^ z.

% commutativity of "or" and "and"
x v y = y v x.
x ^ y = y ^ x.

% distributivity
x ^ (y v z) = (x ^ y) v (x ^ z).
x v (y ^ z) = (x v y) ^ (x v z).

% idempotence
x v x = x.
x ^ x = x.

% absorption
x ^ (x v y) = x.
x v (x ^ y) = x.

% a = false; b = true
x v a = x.
x v b = b.
x ^ b = x.
x ^ a = a.

% negation laws
x ^ (-x) = a.
x v -x = b.
-(-x) = x.

% De Morgan's laws
(-x) ^ (-y) = -(x v y).
(-x) v (-y) = -(x ^ y).

% define * = NAND
x * y = -(x ^ y).

end_of_list.

formulas(goals).

x v y = (x * x) * (y * y).

end_of_list.

And input this into Prover9, which outputs:

============================== PROOF =================================

% Proof 1 at 0.01 (+ 0.00) seconds.
% Length of proof is 9.
% Level of proof is 3.
% Maximum clause weight is 10.
% Given clauses 0.

1 x v y = (x * x) * (y * y) # label(non_clause) # label(goal).  [goal].
11 x v x = x.  [assumption].
21 --x = x.  [assumption].
24 -x v -y = -(x ^ y).  [assumption].
25 -(x ^ y) = -x v -y.  [copy(24),flip(a)].
26 x * y = -(x ^ y).  [assumption].
27 x * y = -x v -y.  [copy(26),rewrite([25(3)])].
28 c1 v c2 != (c1 * c1) * (c2 * c2).  [deny(1)].
29 $F.  [copy(28),rewrite([27(6),11(8),27(8),11(10),27(8),21(6),21(7)]),xx(a)].

============================== end of proof ==========================

Which recovers Rod Carvalho's proof. Note: It didn't actually use most of the Boolean axioms.