Why should a diabetic swab the site of insulin injections with rubbing alcohol first?

Prevent infection. Bacteria live on the skin surface. Alcohol kills them so the needle doesn't pick up living bacteria & deliver them under the skin where they can cause an infection (cellulitis). This is the same reason for an alcohol wipe when blood is drawn from an arm vein. Fingersticks do not generally need alcohol wipes, just a soap & water handwash.
Prevent infection. Skin surface has many different bacteria that typically don't cause infection unless it is introduced into the tissue by the needle of the Insulin injection. Rubbing alcohol reduces the amount of bacteria at the injection site so as to minimize the risk of infection such as a cellulitis. To be most effective, recommend that you let the alcohol dry before injection (it will also sting a lot less).