For example if I just close page or browser or anything else (without clicking "log out" on a site) - can this somehow endanger my account (info leaking or illegal access to an account) ?
When you login to a website, you create a session that is connected to your account. When you logout from the site, you destroy or otherwise invalidate the session. On more security-focused sites, sessions typically timeout after a period of inactivity. It is this session that allows you to access pages in the site without requiring your username and password for every operation.
Sessions can sometimes be stolen or "hijacked". Of course, only an existing session can be hijacked. So logging out ensures that your session won't be hijacked.
Whether you should always logout of sites is something of a matter of personal choice and convenience. I logout from my bank but not from every site.
Note that closing a website in the middle of a transaction can have unpredictable results.