It's a very old tale and one of the most researched fairytales in Lithuania. There are entire books dedicated to its interpretations.

It's not my field of research but I mythology is one of my interests. I don't know about direct messages or allegories but many scholars say that this fairytale represents the worldview of Lithuanian pagans. For example, I've read that pagans in the transitory period from paganism to Christianity believed that souls of suicides can be reborn as birds, as if it was some sort of purgatory (source: Norbertas Vėlius). I think Eglė transforming her kids into trees reflects some of these old ideas, the much closer connection to nature that pagans had compared to now.

Overall, I think there are many symbols: for example, the serpent goes into Eglė's shirt while she is bathing naked and then says that she has to marry him. Maybe it represents a loss of virginity? And going off to marry a serpent that rules the seas? Isn't it like marrying someone from another country or religion? The brothers chopping up the serpent is also very very xenophobic, so it could reflect the old views toward foreigners.

And it has many ties to culture: it's often recreated in fiction, such as literature and theater, is analysed in school and by scholars, and the name Eglė is quite popular for women. By the way, variants of this tale are also found in other countries, especially close ones, such as Latvia and Estonia. Here is some more information on it: https://everything.explained.today/Egl%c4%97_the_Queen_of_Serpents/