Welcome to Maseches Shekalim. Before we turn to the first mishnah of chapter 1, let us set out the basic idea that underlies the entire tractate: the Torah commands that a half-shekel be collected from every male from the age of twenty and up. In the Torah this is mentioned in the context of the census, but the giving of the half-shekel also had a permanent, ongoing role, since the public as a whole - the entire community - bears the responsibility for bringing the offerings in the Beis HaMikdash.
Which offerings come from public funds:
The temidim - two offerings every day, one in the morning and one in the afternoon.
The musafim - the additional offering brought on Shabbos, Rosh Chodesh and Yamim Tovim.
The lechem hapanim and the rest of the communal offerings - every offering that comes at the expense of the public.
All of these had to come from public funds, and those public funds were essentially the half-shekel that each and every person gave on an annual basis.
The public fund was renewed every Rosh Chodesh Nissan. Rosh Chodesh Nissan was the dividing line from which they began bringing all the offerings from the newly collected funds. Our mishnah opens by noting the date on which they began announcing to the public about the collection of the new shekalim, and in the course of that it lists the other matters that took place and were proclaimed at that time.
"B'echad b'Adar mashmi'in al hashekalim":
On the first day of Adar - a full month before the date on which they began taking the funds for the offerings from the new money - they would announce to the public that the time had come to begin bringing the new half-shekels and to contribute toward the offerings of that year.
And not only that, but "v'al hakilayim" - this was also the time when they proclaimed that the hour had arrived to go out and clear the gardens and fields, and to make sure that they contained no kilayim, forbidden mixtures that had begun to sprout.
"Bachamishah asar bo":
From here the mishnah moves on to a different date during the month of Adar, and lists the things that were done on the fifteenth of the month:
"Korin es haMegillah bakerachin" - in cities that were surrounded by a wall from the days of Yehoshua bin Nun, the Megillah is read on the fifteenth.
"Umesaknin es haderachim" - they begin repairing the roads that were washed out during the rainy season, including the main roads used by the pilgrims coming up to Yerushalayim.
"V'es harechovos" - the other types of roads, apart from the major highways.
"V'es mikvaos hamayim" - they begin repairing the mikvaos from everything that accumulated in them over the winter, and also verifying that they are kosher and that they contain the proper measure.
"V'osin kol tzorchei harabim" - they attend to all the needs of the public. The Rishonim explain that this includes the effort of the batei din to adjudicate all monetary matters, and also capital cases and other corporal punishments, as well as all the requirements incumbent upon the beis din:
A sotah whose case needed to be handled.
Redemption of arachin and charamim.
Preparing a new parah adumah.
An eved ivri who wished to have his ear pierced and remain with his master.
Purifying anything connected with a mitzvah that requires purity.
Opening the water reservoirs and the pools of water that had collected over the winter months, so that they would be available to the public during the hot summer months.
"Umetzayenin es hakevaros" - they see to it that the graves are marked. The marking was done with white lime that they would place upon them, so that kohanim and those who are careful about their purity would know to keep away. These markings were somewhat washed away during the winter months, and these too they would renew.
"V'yotz'in af al hakilayim" - on the fifteenth of the month they would send out the agents of the beis din to go and inspect the fields for kilayim, and to see whether the owners had dealt with the matter since it was first proclaimed fifteen days earlier, on the first of Adar. Now they go out and inspect in order to verify that the matter was indeed taken care of.
In summary: In this mishnah we learned that on the first of Adar, a month before the renewal of the public fund on Rosh Chodesh Nissan, they announce about the shekalim and about kilayim, and on the fifteenth of the month they read the Megillah in walled cities and attend to the rest of the public needs - repairing the roads, the streets and the mikvaos, handling all the matters of the beis din, marking the graves, and actually going out to inspect for kilayim.