Why temperature needs an offset
Most unit conversions are a single multiplication, because zero means the same thing in both units — zero metres is zero feet. Temperature scales do not share a zero. Water freezes at 0 °C but at 32 °F, so converting between them needs both a scale factor and a shift.
The 9/5 comes from the size of the degrees: there are 100 Celsius degrees between freezing and boiling but 180 Fahrenheit degrees over the same range, and 180/100 is 9/5. A Fahrenheit degree is a smaller step, which is why Fahrenheit gives finer resolution in whole numbers for everyday weather.
The two scales cross at −40
−40 °C and −40 °F are the same temperature — the one point where the scales meet. It is a useful sanity check: if your conversion does not give −40 for −40, the formula has gone wrong somewhere.
Kelvin has no degree symbol and no negative values. Its zero is absolute zero, −273.15 °C, the point where thermal motion stops. Because a kelvin is the same size as a Celsius degree, converting between them is pure addition. Rankine does the same job for Fahrenheit-degree-sized steps, and Réaumur is a historic scale that puts boiling at 80 degrees — you will meet it mostly in old recipes and nineteenth-century texts.