We probably mostly understand place value for integers.
An understanding of place value, together with the distributive property, is the key to understanding why we mulitply by 10 by just slapping a zero on the end.
Place value for decimals is very similar but we use powers of $1/10$ to the right of the decimal point.
The right way to multiply by 10 is now to slide the decimal point to the right.
We can divide by 10 by sliding the decimal point to the left.
The $20\%$ trick follows exactly from associativity:
$$ \begin{align} 0.2 \times \$22.50 &= (2\times0.1) \times \$22.50 \\ &= 2\times(0.1\times\$22.50) \\ &= 2\times\$2.25 = \$4.50 \end{align} $$The $15\%$ trick follows exactly from distributivity:
$$ \begin{align} 0.15 \times \$22.50 &= (0.1+0.05) \times \$22.50 \\ &= 0.1\times\$22.50 + 0.05\times\$22.50 \\ &= \$2.25 + \$1.13 = \$3.38 \end{align} $$I ran a Google search for "multiplication tricks" and found this page.
All the tricks there can be explained using the algebraic properties of multiplication.
Trick: Double, then double again.
Example: $4\times9$
Explanation: $4\times n = (2\times2)\times n = 2\times(2\times n)$
Trick: Multiply by 10, then cut in half.
Example: $5\times9$
Explanation: $$5\times n = \left(\frac{1}{2}\times10\right)\times n = \frac{1}{2}\times(10\times n)$$
Trick: Multiply by 10, then subtract the number itself.
Example: $9\times13$
Explanation: $9\times n = (10-1)\times n = 10\times n - n$