I understand that both words came from the Italian word bizzarro. However, why their meaning is so different?
According to the Cambridge Dictionary:
Bizarre: adjective. Very strange and unusual.
But, the Diccionario de la Lengua Española states this:
Bizarro: adj. valiente (|| arriesgado).
Is there a particular reason the meaning diverged? At one point did they mean the same in both languages?
In Spanish the word has always meant the same thing. Here you have what the Percival dictionary said about the word in 1591:
Bizarro, brave, gallant.
In 1726 the Royal Spanish Academy said this about the word:
BIZARRO, RA. Generoso, alentado, gallardo, lleno de noble espíritu, lozanía y valor.
Vale tambien lucído, mui galán, espendido y adornado.
So we stick with the meaning of generous, gallant, brave, spendid. And this is the meaning the Academy is trying to keep today, even when the word is most used with the "strange, unusual" meaning (in Spanish, French, Italian and English). So this seems to be a case of "everybody's lost but me", maybe you would like to ask this same question in the English Language & Usage site, or even in the Linguistics site in your question involves more than one language.
Fun fact: there has been traces of the "strange, unusual" meaning in Spanish dictionaries, see the following example:
BIZARRO, RRA. Arq. Edificio de gusto contrario á los principios admitidos en arquitectura. (Zerolo, 1895.)
