This is not stated in EPISODE VI, which is the only mention of Bothans in core canon, but the Expanded Universe used to repeatedly insist that the Bothans were sort of renowned for being spies. Leaving out the obvious observation that it’s a little sketchy to describe an entire race as all being anything, what in particular makes all Bothans automatically suited to be master spies?
"I'm not gonna be PC here, so I'm sure some folks will be offended by this, but all Bothans are spies, am I right?"
In the core story the only thing we know is that Bothan spies managed to retrieve the crucial intel that the new Death Star was not yet operational and the Emperor would be visiting it soon. But here are the three things we know about that:
A) The Death Star was FULLY operational, so that was bad intel.
B) The Emperor wanted the Rebel Alliance to know he would be on the Death Star because the whole thing was a trap, so the Bothan “spies” only managed to get hold of information the Emperor wanted them to have.
C) Many Bothans died to obtain this information, which means they must have gone out of their way to get themselves killed because the Emperor would have needed them alive to deliver the false intel he'd allowed them to obtain.
All in all, the Bothans sound like terrible spies.
But the Expanded Universe gives us some insight into why this might be the case. It could be that they look like this:
"Yeah, a lot of folks think they recognize me. Guess I just got one o' them faces."
The “Legends” (non-canonical) description of Bothans provided in the Wookieepedia is that they are “furry mammalian anthropoids… with canine, feline, and equine features”.
In Heir to the Empire, Zahn described Borsk Fey’lya, the Bothans’ political representative in the New Republic, as having fine cream-colored fur covering his body. This wouldn’t matter much to the STAR WARS universe at large, which is pretty inclusive, but the Empire is repeatedly characterized as being almost exclusively human and highly prejudiced against non-human species. How could a group of cream-fur-covered, horse-faced spies get anywhere near highly sensitive military information unless the Empire wanted them to? And how can the Bothans be great spies if they’re distinctly recognizable as Bothans and have a universal reputation for being spies?
"I'm crushing this spy gig. Nobody suspects a thing..."
The short and simple answer is: They were not good spies. They famously performed the absolute worst act of military espionage in the history of the galaxy, very nearly resulting in the total annihilation of the Rebel Alliance. Which was, I should point out, the exact opposite of their goal.
Nice job, Bothans.