Peter parker was bitten by a radioactive  spider, giving him all of their abilities. Super-strength, super reflexes and.... Precognition? Was a good explanation for the spiday sense ever given or is it one of those suspend your skepticism kind of things?

Bump for longevity.

Suppose it's just feeling when something is coming at you with a rapid speed

I don't know if this count since I'm not the best expert on spiders themselves, but they explained in a show called Spider-Man Tech after the second Spider-Man movie that there are some spiders that have senses that alert them to danger, and that this could be what Peter Parker gained when he was bitten by the spider, another spider ability.

I assume you mean what does Precog have to do with Spiders correct? Not sure about the comics, but in the first film I believe the tour guide gave some kind of brief explanation about a certain type of spider's awareness or nerve reflex conduction sensitivity  being so high that some researches believed it bordered on Precog. I doubt such a spider exists in real life. They do however use their webs as early warning systems, and are highly aware of any movement within said web. Thats all I got :P

He basically can instinctably detect the radio waves, same way sharks detect their pray in the water.
 
here's some more on that

" @Voidheart:  Thats f'n hilarious. "

I've been trying to rationalize the Spider-Sense for a long time, and still can't put my finger on it.

It's kinda like this; close your eyes, and have someone wave their hand really close to your face. It hard for you to explain, but you can feel what they're doing even with your eyes closed. That' s how spider-man spider sense work just with WAY faster speed, and reflexes. 
 Now as for his eyes open; You ever been somewhere when you could just feel something just isn't right? Thats spider-mans spider sense just way stronger.  
 
Did  I make it understandable?   

Real spiders use minute hairs to sense changes in air currents, letting them know if they're about to get smacked by a fast-moving object.

" Real spiders use minute hairs to sense changes in air currents, letting them know if they're about to get smacked by a fast-moving object. "