How realistic is a dictionary attack on a secure remote password protocol (SRP) verifier?
The security goal behind SRP is that an attacker that could either pretend to be a client (and attempt to log into a server that knows the key), pretend to be a server (and allow clients that know the key to attempt to log in), or actively monitor (and modify) the communications between a valid client and a valid server, would learn nothing from an exchange, except possibly whether a single password is valid or not.  This last bit cannot be helped with a password-based authentication method, as the attacker could always take his guess at the password, and then proceed with the protocol with a valid server or client, and see if it succeeds.  The point of SRP is that the attacker can't do any better than that.

Now, in the protocol, the value v is never sent over the wire, and so the attacker does not see it, and so a dictionary attack against that value is impossible (unless the attacker can rederive that value somehow, which is not believed to be possible).

On the other hand, the note you highlighted assumes that the attacker can do more than just work with the protocol; it assumes that he is also able to access the password files on the server.  If he can do that, well, yes, he knows enough to impersonate a server, and he can also perform a dictionary attack on the password; namely for each password P' in his dictionary, he computes:

x' = H(s, P')
v' = g^x'


and check if v == v' (note: it is quite unlikely that you could stumble on a different password that also passes this check)

Note that this is not specific to SRP; for any password-based authentication method, if the attacker can learn everything the server knows, then he can impersonate the server, and in addition, he can perform an undetectable dictionary attack (for example, by simulating a client logging in with various passwords; this is undetectable because since this occurs entirely on the attackers equipment, you aren't informed that someone that someone is trying to log into the system with a long series of passwords).

As for why this is better than a shared secret protocol, well, an attacker can run a dictionary attack even if he didn't have direct access to the server's password files; with SRP, that sort of access is needed.