Support for the claim appears to rest on explicit birth records, although interviews with families in the village were also conducted. The New York Post writes (emphasis mine)
According to birth records, there hasn’t been a boy born there since 2009, though 12 girls have come into the world in that time frame.
I did some digging and was unable to access said records (not wholly surprising, I suppose), or more details on them. It appears that media conducted interviews with families in the area to back this up; the Post's phrasing is ambiguous, but it indicates that boys in general are less common in the town:
Most family interviewed by the press reported having daughters, often more than one.
The Associated Press cites Krystyna Zydziak, the "community head", as saying that ten girls have been born since 2010; the media also talked with various village and local officials, including the county mayor.
Now, the crux of my answer - aside from indicating the likely veracity of the claim - is that none of this should be overly surprising. The village is small (current population 272, according to the New York Times), and therefore there have only been a small number of births in the last decade (12, according to the Post and other media, e.g. Today and Fox News).
If the chance of a baby being a boy and the chance of a baby being a girl are equal, then the odds of 12 consecutive births all yielding girls is (1/2)12 = 0.0244%. That's small, but when you consider that there are many towns and villages in the world, it shouldn't be surprising that at some point, just from randomness, 12 girls are born in a row in a given town. The odds of the same thing happening with 12 boys (which would also be widely reported) are identical, so the odds of 12 consecutive babies having the same sex is 0.0488%. (The Daily Mail notes that in 2017, 207,000 boys were born in Poland, as were 196,000 girls, making the odds slightly lower.)
Of course, more than randomness could be at play; environmental or cultural factors could be responsible. It's possible that maternal diet pre-conception (not post-conception, of course) can play a role. Several studies have claimed that the sex of a child can slightly be influenced by dietary factors:
That said, these results are contentious and not well-supported. I mention them only because the Daily Mail article you mentioned wrote that scientists "have offered to conduct research to investigate the unique situation", and perhaps any environmental factors will be turned up if that research happens.