I can't recall a website that after resetting my password I was automatically logged in, I had to type the password again - which seems quite silly.
From a security perspective, why do websites do that?
From a security point of view, whether the "password reset" logs you in automatically or not is rather neutral. Neither behaviour is inherently more secure than the other. I suppose that most sites do what was simplest to implement on the server side.
A point to consider is that security is achieved as long as everybody cooperates; in particular, the principle of least surprise is of paramount importance. That principle, applied to security, means that trouble is afoot when things don't occur as the human user expects it. In this case, the human user may or may not expect to be logged in after a reset. Let's see what happens if the user is "surprised":
In that sense, a (rather weak) case can be made against logging the user automatically on password reset: security consequences of an unnoticed login can be worse than consequences of an unexpected lack of automatic login.
It's probably just to confirm he chose the right password, what happens if you browse away from the password page on the site?
Normally when you forgot a password a one-time token is generated (often a link) which allows you to change your password and then log in.
If they immediately log you in using the one-time token after which you can change your password, it's a bit weird but not entirely less secure. After all if you can change the password, you could easily log in afterwards anyway.
The important thing is to check if, after using the token, it is invalidated. If the token allows you to change your password over and over again, then there is indeed a security issue.
If you change your password, all of your existing login sessions might be cleared, so that a villain who has your password gets logged out, too.
The fact that you don't get logged in immediately afterwards, or that they don't preserve your current session might just be programmer's lazyness, or what Lucas said.