It provides an easier way to save and use your passwords to all the internet sites / accounts that you have. Source
It gives you an account that then houses all your individual passwords within, this password manager account then allows you to access all the individual passwords by only having to submit your single password manager account password. Source
In general it is more safe, because most password managers use two factor authentication, however if your password manager password is user-selected the entire system is only as strong as that password. Source
LastPass, Keeper, Dashlane, 1Password and lots more Source
I honestly don't cause I have a pretty good password "system" that I use where no 2 passwords are the same for me. I've always been concerned when my apple device tries to get me to use a suggested password because I think that when I would eventually get rid of that device I would have to put them all back in, and most likely not remember them, and therefor have to change every one of my passwords. However, after learning what I have I'm a bit more intrigued to look into them.