Why do shippers choose a particular ship? Because shipping typically involves couples, I will assume that a given "ship" is a couple. And because this is largely a broadcast media show forum, I will restrict myself to broadcast shows.
In my experience, shippers tend to believe in a given ship for one of two reasons: 1. they identify with one character very strongly (almost always of the same gender) and identify within the show another character that reflects the person's most desirable characteristics in a potential mate and they like what the pairing represents within the context of the show; 2. they identify with what the paring represents within the context of the show and then spot character traits that they admire within the pairing. These character traits could be style, personality, interests, attitude, or anything else. The shipping may be an outgrowth of the person's interest in the show or the person becomes interested in the show because of the ship.
Moreover, shippers tend to idealise the characters within their ship and their relationship itself. The most common identifications for the shippers are a failed past romance, a future romance with someone the person is not dating, or an external idealised relationship. Thus, you are typically are identifying with a character and a situation you probably had or wish you had with someone else; this will cause you to idealise one or both sides of the ship. It is very important that you recognise this combination of identification and idealisation; otherwise, your shipping could be come pathological. PM me if you want to know what "pathological" looks like.
Again, remember that you can expect those who support an opposing ship have the same foibles as you. If you believe that people who support an opposing ship have blind spots or bizarre rationalisations for the ship, then remember that you have them too. So you should be very circumspect before criticising the intelligence or character of those who believe in a competing ship. Expect that any ship warring will get taken personally.
Next: Example - iCarly.