Roots, Wurzeln and rādīcēs all go back to the same Proto-Indo-European (PIE) root: *u̯r(e)h₂d- (or something like that).
English root is borrowed from Scandinavian. In Old Icelandic it was spelled rót, in Modern Swedish, for example, rot (pronounced like English root).
Because the initial consonant cluster of PIE *u̯r- was hard to pronounce, *u̯- (PIE spelling convention for w) was dropped, so we’re left with *r + *h₂ (that develops into a vowel) + *d (that develops into *t, Grimm’s law)
German didn’t drop the initial *u̯-, instead it developed a vowel to make the word easier to pronounce: *wur-. The laryngeal h₂ was consequently dropped, and Grimm’s law took care of PIE *d which became Germanic *t. The result is *wurti- ‚root‘. This was combined with the word for ‚rod‘, *wal- (thus designating the main root, but subsequently roots in general), resulting in Modern German Wurzel. Würzen ‚to season‘ is another descendant of *wurti-, which also meant ‚herb‘.
Latin rādīx, inspiring this blog’s name (rādīcēs is the plural of rādīx), continues *u̯r(e)h₂d- as well. It is also the base of radical and radish.