yes and no.
What last pass is great at doing is creating a virtual storage of your passwords that is secure. This allows you to confidently create passwords that would be ridiculously hard to remember without writing them down somewhere while at the same time allowing you to have different strong passwords for every site you log into with no need to remember them.
This means if one site is compromised then all of your other sites wont be compromised as well. Most people use the same password in multiple places so if one place falls then the hackers have gained access to multiple sites. Creating different passwords for every site you log into is hard to keep track of without some sort of password manager.
You could certainly rely on the browsers ability to save logins but those are not nearly as secure as last pass that encrypts everything by default.
You could certainly just use last pass to generate strong passwords then write them down, or you can even export them to a file you could store somewhere safe.
If you access a site from a machine that does not have last pass on it you would just have to know the password you used for that site.
What I do is use chrome and I sign into chrome with my google account. When you do that Chrome retains all of your settings, bookmarks and extensions, including last pass. So when I go to another machine as long as it has chrome on it all i have to do is sign into chrome on that machine and last pass will be installed automatically. Then all I have to do is sign in with my last pass account and all of my passwords are there ready to use. When I leave that machine I sign out of chrome and it is gone.
Alternatively you can sign into the last pass website and retrieve your password there from a machine that last pass was not installed on.
If you use a browser to save your passwords it is trivial for someone that has access to that machine to access them. if you use last pass and don't leave your instance of last pass signed in it becomes nearly impossible to do so.
On my home machine I leave last pass signed in. Not the safest practice but I work on computers for a living and keep my machine clean of infections and the machine itself is passworded. If someone were to steal the machine they could get to any passwords that were stored in my browser but none of the passwords stored in last pass.
You certainly don't have to use last pass. It is simply a great way to use different passwords for every site you visit without having to remember every one while also keeping them secure( I don't know any of mine other than my gmail and my last pass). Also assuming you set up your account on whatever site with a recovery email address you can always recover your password or reset it as long as you have access to your email.
In my case if I lost access to last pass for some reason. I would have to go to each site and recover my passwords one by one.
Hopefully that all makes sense but in case it didn't here is an article from a professional that explains the benefits.
http://www.howtogeek.com/141500/why-you-should-use-a-password-manager-and-how-to-get-started/
= new reply since forum marked as read