The Ikarim where 13 principles during the time of Rambam, that could be used to differentiate Judaism from Christianity, Islam and other popular beliefs at the time. During the Rambam's time, religious wars were gaining a status of their own, and the Jewish people were being seen as a religion rather than a nation in exile. These Ikarim were needed to help the Jewish people feel pride and distinguish themselves in the eyes of the gentiles, while also being compatible with them. Even though Rambam did not live in Christian countries, he still had contact with those Jews, and knew his works would reach them.
An Explanation of how this works with the 13 Ikkarim: (using the translation from your link)
God exists; God is perfect in every way, eternal, and the cause of all that exists. All other beings depend upon God for their existence.
This establishes the foundation of the religion as being monotheistic, and in line with Christianity and Islam. A point they can agree on.
God has absolute and unparalleled unity.
This is to separate Judaism from Christianity at the time.
God is incorporeal--without a body.
This is to give commonality between Islam and Judaism so that Jews are not regarded as a lower status than they already were.
God existed prior to all else. (In a later version of the Thirteen Principles, Maimonides included the notion that God created the world from nothing [creation ex nihilo].)
This is in objection to the beliefs of the philosophers.
God should be the only object of worship and praise. One should not appeal to intermediaries, but should pray directly to God.
This is to be in contrast to Christianity and superstitions of his day.
Prophets and prophecy exist.
This is used against Philosophers, to give the basis of the Torah for being the correct books to derive from.
Moses was the greatest prophet who ever lived. No prophet who lived or will live could comprehend God more than Moses.
This is in contrast to both Christianity and Islam who say the Commandment's of Moses could be reneged.
The Torah is from heaven. The Torah we have today is the Torah that God gave to Moses at Sinai.
This is to give the Jews confidence in the words and laws of Moshe. Perhaps one might think that Moshe was the best prophet, but we have remembered what he wrote wrongly. This was also an argument against the Kaarites, applied to the Talmud and oral laws.
The Torah will never be abrogated, nothing will be added to it or subtracted from it; God will never give another Law.
Again, directly against Christianity and Islam.
God knows the actions of humans and is not neglectful of them.
This would is required to help the Jews feel confident that perfoming mitzvot is not for naught. Also, to prevent people from saying that their intellect is refined but their actions don't have to be.
God rewards those who obey the commands of the Torah and punishes those who violate its prohibitions.
Perhaps someone might want to leave the Jewish community and only keep Judaism in private. This resist the claim that they can "blend in" with the gentiles and all will be ok.
The days of the Messiah will come.
This gives the Jewish people the strength to survive in the times of hardship and persecution. It's also an argument against Christianity.
The dead will be resurrected.
Also an argument against Christianity, and a statement that helps Jews remain Jewish.