Interfere comes to us from the French entre– (“between”) and ferir (“to hit”). Interfering with something is really just hitting it right in the middle of it, breaking it up! Ferir comes from the Latin, for the same, Ferire.
Curiously, Ferire evolved into Spanish Herir — the Initial “F” turning into a “H”. It turns out, this is a common pattern as Latin evolved into Spanish — but in no other language! Just look at Filial and Hijo, or File and Hilo, or Fig and Higo.
Thus, the h-r of herir maps to the (int)-f-r of interfere.