The Gemara discussed the importance of mentioning our exodus from Egypt both in the morning and evening. It was determined that the chapter on Tzitzit was the most appropriate because it both included Egypt and had other important messages as well.
In the Tzitzit paragraph, aside from Egypt and, of course, the commandment to put fringes on four cornered garments, there are other laws to go with it.
There is a Mitzva of לא תתורו אחרי לבבכם ואחרי עיניכם, not to turn after our hearts and eyes. This refers to the prohibition of reading heretical publications that might turn our hearts away from Hashem. It also includes the prohibition of having lustful thoughts in the way one may look at women. We are to guard our eyes from such thoughts that could lead to forbidden actions.
The final idea is the acceptance of the yoke of the Mitzvot, in general. This is learned from the words, למען תזכרו, in order that you remember.
Although there are other places in the Torah where Egypt is mentioned, it is understandable why ציצית was incorporated to be part of the Shema.