However the Mechilta sounds a bit different. ה' נתן לכם השבת - שתשמרוה . This doesn't sound like the לכם we had in mind...
I believe there's an amazing insight here. How or why does the medrash learn שמירה from לכם ?
I would like to suggest, how does shabbos become ours? If Hashem is giving it to us as a gift, what קנין - act of acquisition is there for it to become לכם ?
This is what the medrash is telling us! The acquisition of shabbos is שתשמריהו! Shmirah does not mean to KEEP shabbos, it means to SAFEGUARD shabbos. When one accepts the responsibility of shmirah upon something, it becomes theirs to some extent. We find this by chometz on pesach and in general we know that the responsibility over the item takes effect with a kinyan that establishes the relationship between the receipient and the item.
The medrash is telling us that perhaps keeping the laws of shabbos is ok and renders you a kosher observant Jew, but shabbos only becomes YOURS as a gift from Hashem when you not only keep it but when you accept to SAFEGUARD it. Not only from your own desecration but also from others!
May I ads that this passuk actually specifies o ly one aspect of shabbos to safeguard - הוצאה. Some learn this passuk as hinting to the melacha of transferring objects and some say it refers to Techum Shabbos. Both are uniquely viewed as being the 'lightest' desecrations of shabbos.
This is what is meant by safeguarding Shabbos. Not only to prevent coarse obvious chilul, but keeping Shabbos safe from any breach.
We have this opportunity no in our community. Not only to observe Shabbos for ourselves but to care and try to promote its sanctity, safeguarding it from the sadly common desecration of carrying on shabbos.