To "keep" it (שמר, "shamar", literally, "to guard") means to follow the laws of Shabbat. Someone who follows the laws of Shabbat is called "shomer Shabbat". The basic meaning of שמר is guarding something -- the same root is used for a bodyguard, a security guard, a military guard, a prison guard. So, it really means, "to guard the Sabbath", and it refers to actions that you do and do not do.

To "remember" it (זחר, I think, although I could have spelled that wrong, "zachar", "to remember") is to bring forth in one's mind the reasons for Shabbat, and to understand it. It is mental and spiritual.

So, by having the two commandments, we are commanded both to obey the laws with our body, by taking and refraining from actions, and to understand and feel the laws with our mind and soul, by thinking and feeling.