On each day of the week in the times of the Bet HaMikdash, the Levi’im would recite a psalm corresponding to that day. To commemorate this, a different Mizmor is recited on each day of the week during the Shaharit prayer, known as the Shir Shel Yom. Before each Mizmor, an introductory phrase is recited: “Hashir Shehayu Halvi’im Omrim Al Haduchan Beyom [day of the week] BeShabbat” (lit. “the psalm that the Levites would recite on the platform on the [day of the week] from the Sabbath”).
The reason the psalm for a particular day is prefaced with that sentence can be explained as follows: The Mechilta d’Rabbi Yishmael (Parashat Yitro, Parasha 7) says that one way of fulfilling the commandment to remember the Sabbath by doing things in anticipation of Shabbat, such as preparing a delicious dish for the Shabbat meal. The Ramban (Shemot 20:7), basing himself on the Mechilta, explains that remembering and thinking about Shabbat is a constant Mitzvah. As such, the Arizal (Sha’ar HaKavanot 60a) that when one prefaces the daily psalm by counting the day of the week relative to Shabbat, one is fulfilling the commandment to constantly remember Shabbat. Rabbi Yosef Hazan (Shu”t Hikre Lev, vol. 2, Orah Haim, § 32) that this introduction is a significant custom and that when reciting it, one should have in mind that one is fulfilling the commandment of “Zachor Et Yom HaShabbat Lekadesho”, to sanctify the Sabbath by remembering it.
Summary: One should have in mind that, by reciting the introductory sentence before each Shir Shel Yom, one is fulfilling the commandment to remember Shabbat.