You're not giving Dropbox your password. You're giving it to your operating system. Dropbox is doing something that requires elevated privileges and your operating system is saying "hold up a sec, you need permission." Your operating system then prompts you for your password to allow Dropbox to do whatever it is that it needs to do. Your password is never given to Dropbox.
As for what Dropbox is trying to do... there's no way to know exactly, but it's usually going to be something like fixing/setting overlay icons, updating, adding or changing services that are needed, etc. Simply put, it's nothing nefarious.
Do you work or study in the creative industry? If so, which area?