Dowries seemingly are completely one sided. The husband's family gets both the bride as well as the dowry, so what's in it for the bride's family?

At least in Classical Athens, it was that you are no longer responsible for the bride. As Isaeus writes in On the Estate of Pyrrhus:

For what was likely to be the good to him of the marriage, if the husband could dismiss the wife whenever he wished? And this he certainly could do, if he had made no stipulation that he should receive a dowry with her.

Essentially, in Classical Athens, an Athenian woman was not a citizen in the sense that she had full civic rights. She always had to have a male guardian, who was responsible for looking after her. When she was born until she married, that would be her father, or if he was dead, her brothers.

The dowry ensured that when a woman married, she stayed married because the dowry continued to be hers. If there was a divorce, the husband would have to pay back the dowry, and the fact that there was a record of the dowry being paid meant that he couldn't just abandon her.

Basically, a dowry was a way of a) persuading a suitable man to marry your daughter (especially because, at least among well off Athenians, Athenian women weren't meant to even leave the house except on certain special occasions, usually religious in nature, and, as Winkler writes, to even mention a citizen wife by name was seen as a "shame and an insult"; in Lysias III the speaker says of his nieces that "their lives have been so well ordered that they are ashamed to be seen even by their kinsmen"), and b) ensuring that that marriage was going to last, by both making a legal record of it, and having an economic incentive for it to continue.