How does this proof of Theorem 1 in Spivak's Calculus work?

The lines are read left to right and top to bottom. So the symbol ≤ on line 2 relates the last expression of line 1 to the next expression on line 2. Similarly, the symbol = on line 3 relates the last expression of line 2 to the next expression on line 3. The same thing could be written in one line as follows: \begin{equation} \begin{split} (|a+b|^2) = (a+b)^2 &= a^2+2ab+b^2 \leq a^2+2|a|\times|b|+b^2 = |a^2|+2|a|\times|b|+|b^2| = (|a|+|b|)^2, \end{split} \end{equation} but it is easier on the eyes to write each new expression on a new line.

For any number $x \in \mathbb{R}: x \leq |x|$

Hence, $ab \leq |ab| = |a||b| \implies 2ab \leq 2|a||b| \implies a^2 + 2ab + b^2 \leq a^2 + 2 |a||b| + b ^2$