It's probably stubble or hair follicles below the skin that your razor can't reach. That's common with dark, coarse hair, especially if you have fair skin. Waxing rather than shaving is one solution, since it removes the whole hair follicle. For underarms, I recommend hot wax rather than wax that you use strips with. It's more difficult to use but works well on coarse, deeply rooted hairs. Or visit a salon. Keep in mind that waxing can be very painful and irritating at first, but as you do it more, it gets less painful and your skin begins to respond better to it. If waxing just isn't for you, you can try a cream that dissolves hair below the surface, but beware since they tend to be irritating and underarms are sensitive. You can also try a better razor or a different shaving cream. Otherwise, jus don't worry about it. No one will be focusing too much on your armpits! You probably notice the spots more than anyone else ever will.
I think that is because the skin in that area is thin and so you can see the roots of the hair. I get this too, usually the longer I wait in between shaving the less noticeable it is, also try getting a closer shave by turning on the cold water so the hairs stick up and use a sharp/new razor