In the Gemara, the translation of שקל is to take. The Chidushei HaRim zt'l said that on Shabbos Shekalim, the zman is mesugal for one to take himself with both hands (Imrei Yehudah). In other words, it is a time for change. It is a time to take himself in his hands and begin anew.
Rebbe Pinchas Koritzer zt'l looked forward to this time of year when we read the four parshiyos. "I want to meet with the Eibishter," he said (Imrei Pinchas 4:225). This is as we say in the yotzros of Musaf on Shabbos Shekalim, נשא אדון עלינו פניך אור, we request that Hashem shine the light of His countenance on us. It is a time when we can keviyachol meet with Hashem. The Ruzhiner zt'l said, "In the [Shabbosim of the] four parshiyos, each Yid can become attached to the Shechinah because each week represents another letter of Hashem's name (ה"הוי)."
In Megillas Esther it states (Esther 7:8) המן ופני חפו, "Haman's face was covered." The Alshich HaKadosh explains that the ancient law of Persia (and of many other countries of old) was that if a person convicted and sentenced to death saw the king, he was pardoned. When Achashveirosh's anger was aroused on Haman, someone quickly covered Haman's face, so Haman wouldn’t look at the king’s face and be pardoned. As we said above, during the weeks of the “four parshiyos” we merit seeing Hashem’s face, keviyachol. As we say on Shabbos Shekalim, אדון עלינו פניך אור נשא. And when one sees the King's face, all his sins are pardoned.
Furthermore, it states (30:13), העובר כל יתנו זה ...הפקודים על. The Yerushalmi writes that כל הפקודים על העובר means "all those who transgressed Hashem's commands" (and sinned with the egel), with the machatzis hashekel, על לכפר נפשותיכם, they will have atonement. Similarly, we attain atonement on this Shabbos.
The Sfas Emes (Shekalim ג"תרל) explains that this atonement is also applicable in our times because we desire to give the machatzis hashekel, and that is sufficient to attain atonement.