The explanation is one of Purim's core characteristics: hidden provenance. The name of the scroll, Esther, means to hide and everyone puts on a crazy costume to hide themselves too. It's the festival of the hiding, and that's why Hashem's name isn't mentioned in Megillat Ester.
However when writing the actual Megilla, many scribes felt the need to highlight parts where the Megilla was referring to Hashem. In fact, that's the reason why the so-called Hamelech Megillot - which are arranged to have the word Hamelech at the beggining of every column - are so popular and command higher prices in the Stam marketplace. The word Hamelech is written numerous times in the Megilla starting every column with this word would be like the common Jewish practice of writing BSD (besiata dishmaya - with God's help) at the top of a note or page, since Hamelech can also be understood as a reference to God.
(Similarly, I've once heard that this is why every Talmud page has always four long lines of Rashi and Tosfot at the very top. It is presumably a hint to the four-letter name of God, Yud-Kei-Vav-Key. )
Be it as it may, scribes don't only use the Hamelech layout to highlight Hashem in the Megilla. Below is a 200 year old Megilla where the scribe wrote the four-letter of Hashem in big font (it is actually reversed).
In this other Megilla (hat tip to Melech), the Sofer did something similar but instead of writing big letters he made three dots on top of the corresponding letters:
I think this was a better strategy because there's a Mesorah that dictates which letters should be big (and small) throughout Tanach. It's not optimal to start inventing your own (that's the reason behind the 11-lined Megilla of the Gaon - see here for full post on that), but again, the concept is the same.
Many people attribute this practice of highlighting the Shem in the Megilla to the Hacham Tzvi, who was perhaps the most respected Ashkenazi (don't be fooled by the name) Rabbi of the 17th century Europe. I looked extensively for this source and was pointed to the Siddur of the Yaavetz , which you can read below. The Yaavetz says in the name of the Ari that we should be Mechaven (meditate) in the special Rashei and Sofei Tevot in the Megilla which are references to the name of God. Most likely, scribes decided to enlarge (or dot) these hidden names of God to make it easier for us when reading from the Klaf and thus be Mekaven.
Be it as it may, the Hamelech layout, large letters and dots are all well-established Minhagim that enhance and make our Megillot look even nicer.
Happy Purim and drink very responsibly