Answer

I don’t think it has any special significance. It is an opportunity to connect to the world of the mystical, since it is the day of Rashbi’s passing. The meanings attached to this day were read into it after the fact. See Hatam Sofer, Yoreh De'ah, responsum 233:
Indeed, I know, for I have heard that now worthy generations are being prepared, and from afar they come to seek God in the holy city of Safed on Lag BaOmer, at the celebration of the late Rashbi. And although all their intention is for the sake of Heaven, their reward is certainly great… nevertheless, for that very reason I would be among those who separate themselves, like ben Dorotai, so that I would not have to sit there and alter their practice in front of them, and so that I would not wish to join them in this. For the Pri Chadash already argued extensively in Orach Chayim 496, in his pamphlet on prohibited customs, section 14, about places that make a festival on the day a miracle happened to them… that in my humble opinion we say this from an a fortiori argument: if from slavery to freedom we say song, then from death to life all the more so. But to establish a festival on a day on which no miracle occurred, and which was not mentioned anywhere in the Talmud or the halakhic decisors, not in any place, hint, or allusion—only refraining from eulogy and fasting is a mere custom, and I myself do not know its reason.
And in the prayer book of Rabbi Yaakov Emden it is written, based on the mystical tradition, that it is like a court in which all are liable, and yet it comes out innocent—that is, “splendor within splendor.” See there. But according to this, it would have been fitting to establish every good thing when we reach the powers, namely “strength within strength,” on the ninth day of the Omer; except that in any case those are the days of Nisan, when eulogies are not delivered…
As for what the Chida wrote at the end, there is much to elaborate on, but this is not the place.
——————————————————————————————
Questioner:
Why does Hatam Sofer write that he does not know the reason why Tachanun is not said on Lag BaOmer? Does it really have no source at all in the revealed, non-mystical tradition? After all, the Shulchan Arukh and the Rema, if I remember correctly, also bring this as Jewish law, and if so there must be, I assume, medieval authorities who ruled this way or instituted it—so I don’t understand how it can be that Tachanun is not said on that day without a source.
——————————————————————————————
Rabbi:
There are customs that develop, and the halakhic decisors do not know their source or rationale. I’m not sure Hatam Sofer means to say there is no reason, only that he does not know the reason. See my article: A Homily for Lag BaOmer.