How is a type 1 error more dangerous than a type 2 error in statistics?
I would say it's not always more "dangerous". That perception might be from two things: 1) in many societies, it is considered to be worse to convict an innocent person that to acquit a guilty person and 2) we tend to want to give the null hypothesis the benefit of the doubt, unless there is strong evidence against it.
When considering a test of a null hypothesis
In a courtroom, the null hypothesis is that the defendant is innocent while the alternative hypothesis is the opposite conclusion, that the defendant is guilty. Therefore, a type 1 error in this context is the conclusion that the defendant is guilty when in fact the person is innocent; and a type 2 error in this context is the conclusion that the defendant is innocent (or at least that there's not enough evidence to convict), when in fact the person is guilty.
Both kinds of errors are "bad", though societies typically think of the first kind of error as worse. The second kind of error here is more "dangerous" to the society, as it could result in letting a violent criminal go free.
In business, either kind of error can be "bad" or "dangerous". For example, if your company makes cars and you have tried to make some aspect of a crash test come out better (safer), the null hypothesis would be that the change makes no improvement to safety (or even makes safety worse) while the alternative hypothesis would be that the change does make an improvement to safety. In this context, a type 1 error would be the mistaken belief that the change has improved safety, when in fact it has not (so this could lead to more people dying and perhaps lawsuits). And a type 2 error would be the mistaken belief that the change has made no improvement when in fact it has (so this would mean a "missed opportunity" to improve safety, which could also lead to more people dying than they would otherwise).