Where comes the word root when talking about the points when a polynomial gets the value $0$?
The usage of "root" is so old that it is hard to pin-point the first usage and as to why we call them roots. At in English, the usage dates back to 14th century, and if this "root" concept was borrowed from Arabic, as suggested by Jeff Miller of the "Earliest Known Uses of Some of the Words of Mathematics", then it will be even more difficult to rationalize why a root is called a root. Google Translate also shows that the Arabic word for root (of a polynomial) is jazr, which literally means a root.
According to the Earliest known uses of the word of mathematics
RADIX, ROOT, UNKNOWN, SQUARE ROOT. Late Latin writers used res for the unknown. This was translated as cosa in Italian, and the early Italian writers called algebra the Regola de la Cosa, whence the German Die Coss and the English cossike arte (Smith vol. 2, page 392).
Other Latin terms used in the Middle Ages for the uknown quantity and its square were radix, res, and census. The term root was used by al-Khowarizmi; the word is rendered radix in Robert of Chester’s Latin translation of the algebra of al-Khowarizmi. Radix also is used in translations from Arabic to Latin by John of Seville, Gerard of Cremona, and Leonardo of Pisa. For an early English use of root, see addition.