Why are "roots" so unrelated to derived words?

Sometimes the changes between the root in a parent language (in your example, the reconstructed Proto-Indo-European) and the reflexes in different daughter languages is minimal and sometimes quite extensive (your example of PIE *bh- > L f-). For what it's worth, PIE *bh- stayed pretty much the same in Sanskrit, cf. bhrātṛ 'brother'. There are even some strangeer changes closer to our time in Romance languages (where we hav the parent language, Latin): e.g., in Ligurian dialects of Italian, initial ClV-, which in Standard Italian goes to a palatalized CjV-, becomes instead an affricate. L. plus ciu, VL *blancus 'white' > Lig. gianku; cf. Standard Italian piu and bianco.

The relationship between a PIE root and one of its daughter languages may seem arbitrary or tenuous, but after you studied historical phonology for a while, it starts to make sense. You can cannot predict how a language will develop out its parent,m but given enough time you can understand how a sound can change over a couple of thousand years from one that seems very different indeed.

Ceci n'est pas un seing.