Here are some interesting opinions unofficial:
1. That was the color of the spider who bit him in the movie. As far as the comic books go, those were just the colors he chose.
2. Most superhero costumes in the 1960s favored bright primary colors such as red and blue, sometimes with yellow. (Recall Superman, the Flash, Iron Man, Captain America, Captain Marvel, the X-Men, etc.) Exceptions include Green Lantern and Green Arrow. By contrast, villains often sported purple and/or green, along with gray and black. (Dr. Doom, Magneto, Dr. Octopus, the Lizard, Kang, Green Goblin, Luthor-as-supervillain, Impossible Man, etc). Batman stands out as a major exception because he uses the same black/gray motif common among villains, suggesting his "darker" nature compared to other superheroes. Likewise, Marvel broke the four-color barrier by introducing the green Hulk, who always sported purple pants--another departure from the standard superhero motif.
3. Because Superman Costume is red and blue! Spider-Man and Superman are both just average joes behind the super scenes. Marvel used these colors as an allegory to a superhero with similar circumstances.
4. Superman was a DC comic character, and spiderman was Marvel, different comics, also, back when comics where most popular, the creators stressed a lot of patriotism in their characters, using red blue and some white
6. Because the colors of the American Flag are Red, White and Blue. Spider-man’s suit is Red and Blue plus his eyes are White.