Why is it important to remember that Hashem delivered us from the slavery in Egypt?
Rashi: Because when He did so, He acquired us on condition that we become slaves to Him and that we are therefore bound to observe His Mitzvos. 1
See Sifsei Chachamim.
What are the connotations of "Al-Kein Tzivcha Hashem Elokecha La'asos es Yom ha'Shabbos"?
Ramban #1 (citing Ibn Ezra): 'That is why Hashem commanded you to allow your Avadim to rest'. 1
Ramban #2 (citing Moreh Nevuchim): Hashem commanded us here to keep the Shabbos to remind us of the Chesed Hashem, who redeemed us from Egypt, where we slaved day and night without respite, whereas in the first Dibros 2 in Yisro, Hashem commanded us to keep the Shabbos to remind us that He created the world in six days and 'rested' on the seventh. 3
Ramban #3: On the one hand, keeping Shabbos reminds us of the miracles of Yetzi'as Mitzrayim 4 that Hashem took us out of Egypt; whilst, conversely, the miracles of Yetzi'as Mitzrayim remind anyone who doubts the Creation that Hashem did indeed create the world 5 and that He retains connections with it and that He is all-capable.
See Ramban's objection to this explanation.
Ramban: And the two are independent issues which are unconnected.
The Ramban objects to this explanation as well, seeing as not working on Shabbos will not really remind anyone of the slavery in Egypt.
Since how could He have otherwise change the laws of nature if He did not create them.