Salut!
One reason why someone might wish to write to a magazine is that the respective person feels he/she is gifted and would try his/her chance to have an original article published in that magazine. For instance, if the person can easily write political articles, literary assays, or even interesting recipes for specific magazines, he/she will choose to write to a magazine's editor, hoping the article will be read and published.
Another reason would be when a person has an opinion about an article which was already published. The reader can express his/her viewpoint on someone else's article.
A third reason why somebody might write to a magazine is to answer to a call for a competition organized by a certain magazine. Looking for new talents, the editors of a magazine may intiate competitions for young gifted freelance writers. Of course, in this case, the young people looking for a successful career will write to the magazine, to express their will to give it a try.
