I don't trust any password management site. Why would I want all my passwords kept on some third party server? You're better off hiding a written note in your home. What if someone just attacks that server instead, then they have all your passwords instead of just one of them.
Not sure if you're talking about lastpass, or if you thought keepass2 was a web-based password management app buttttttttttt I'll try and ease your mind a little/enlighten you...
As long as the database in which your password(s) are stored on any given web server, even if that server is compromised, your passwords (in the environment of a properly managed database) will be secure. Part of this is your responsibility to make a strong password, no realistic amount of help is going to fix a weak password.
In simple terms this is what is going to happen: you give mywebsite.com your user name/password, they shouldn't need that information for anything other than authentication, therefore they will hash the password (preferably with some sort of salt), and store the hashed value of your password. When you authenticate in the future, they only compare the hash your browser sends them, to the hash they have stored - if they match, congrats you entered the same password and you're let in. Without going into the details of attacking something like this, suffice it to say that if your password is weak and the database is compromised, it will be trivial to crack your password. But if it strong, odds are your password is safe from anyone other than a state-backed actor.
Not to mention that it'll be exponentially more difficult to attack a properly hardened webserver than most people's local machines/home networks.
Now... I haven't done my research on lastpass because I don't use it but I'd be willing to bet that your passwords are safe there...
As for KeePass - all the passwords are only ever stored locally on your machine. You set up a master password and have the option of adding a key file and OS user account as additional layers of authentication to access your password database. You then get to set up your list of passwords, which KeePass will randomly generate for you if you'd like (yay copy/pasta) and never have to worry about remembering anything other than 1) your master password and 2) the location of your key file.
In order to compromise this an attacker would need full access to your local machine, they'd have to know where your key file is located, and they'd have to be logged in as you (well I guess that part is easy if your Windows/Linux/OS X password sucks). Anyways, the point being... the odds of this happening are incredibly low unless you derp up.
Edit: https://blog.lastpass.com/2010/07/lastpass-gets-green-light-from-security.html/
^ If lastpass is good enough for Steve Gibson, it's good enough for me. I'll stick with KeePass because I already have it set up, but there's some more reading material for those interested.