Why does "quadratic" describe second power when "quad" means "four"?

Because quadratus is the Latin for "square" due to there being four sides on a square.

The second power of a number is called its square because if we have an integer, and construct a square with that number of items on each side, the total number will be its second power. E.g. a 4×4 square having 16 items:

* * * *
* * * *
* * * *
* * * *

It is also the case that the area of a square with a real number length side, can be measured in terms of the square of that number.

Related language trivia: In the middle of the 16th century, while squared was being used in this sense by some, it had yet to become current, and there was a brief use of zenzic, which means the same thing (via a Latinised, Germanicised, Italian translation from Arabic; it's a convoluted etymology). This form was productive so zenzizenzic meant the 4th power (squared square), zenzicube the 6th (squared cube) and for the 8th power we have zenzizenzizenzic, which is often claimed as the English word with the most Zs in it. It's a doubtful claim though, first there's nothing stopping us from using zenzizenzizenzizenzic to have even more, and second there seems to be about nobody using it any later than 1560 other than in sentences containing the phrase "has the most Zs".

Because the second power is a square, and a square has four sides. In fact the word square itself comes from quad-, too.

Quadri-, quadru-, quadr- is a Latin prefix related to the verb quadrare ‘to make square’, quadratus ‘square’ etc. Quad is simply an English shortening of quadrangle, quadruplet etc. So quadri- and quad are actually the same word.