Rosie, to put your mind at ease I'm going to jump in here and say that your confusion and your criticisms are quite appropriate. The author's explanations/excuses make no sense at all, and reflect very poorly on him as a service provider.
Here's the situation; it's a bit long-winded, but if you bear with me I'm sure we can solve your problem.
As you suggested, the problem is SMTP authentication. For anyone who may not be familiar with the term here is a brief if simplistic explanation:
We all know that spam is the scourge of e-mail users. One way that spammers spread their garbage is to route it through poorly configured computers that do not belong to them and to which they should not have access. Machines that have been taken over in this way are called 'open relays'. Basically that means they are open to abuse because no authentication (username/password) is required to send e-mail through them.
For several years now there has been a global effort to eliminate open relays. One way to rectify the situation is to require valid authentication of anyone trying to send mail through the machine. That means configuring the computer to require a valid username and password.
While annoying to all of us, spam is a particular problem for operators of mail servers, not least because all of those millions of rubbish messages consume resources and anger customers. So, quite understandably, responsible Hosts have implemented SMTP authentication on their servers.
In other words, to send e-mail through their mail server you must have a valid account on it and know the username and password. The username and password must be sent with (or in association with) the e-mail message, or the mail server will not process it.
Now, while there are still plenty of mail servers that do not require SMTP authentication, their numbers are shrinking all the time. Day by day, more and more mailservers require SMTP authentication.
So what is the future of software that attempts to send e-mail but does not provide a means of sending SMTP authentication with a message? The obvious consequence is that more and more installations of such software will begin to fail.
What then is the consequence for site owners who use products like "Contact Form 7"?
Very often the contact form is the only means that a site visitor has to get a message to the site owner, so if the contact form suddenly stops sending messages (for the reasons outlined above) the site owner may never know about it. If it's a business-related site such a failure could be critical and costly. Which obviously is just not good enough.
Unfortunately plug-in author Takayuki Miyoshi has done an abysmal job of explaining this and the solution.
Yes, there is a solution.
I don't want to get into the intricacies of the code that drives e-mail sending from a web form, so I'll just state the problem without further explanation. "Contact Form 7" appears to use PHP's wp_mail() function which, as written, won't provide the necessary SMTP authentication. The answer is to employ another plug-in that forces all e-mail to be sent by SMTP instead of the PHP mail() function. DISCLAIMER: I'm not a PHP programmer but I think that's a fair assessment, while trying to keep things simple.
So if you want to use "Contact Form 7" and if your mail server requires SMTP authentication, then you need to install a plug-in such as "WP Mail SMTP". All it does is provide a configuration page where you can insert your username/password and other relevant details. The one quirk with "WP Mail SMTP" is that it will appear in your WordPress Settings list as simply "Email". Why these guys won't use Settings names that we expect to see is beyond me. You searched for, downloaded and installed "WP Mail SMTP" so isn't it reasonable that's what you'll be looking for in the Settings list? And it can be a long, unordered list if you have a lot of plug-ins installed.
If you are not worn out by now, there is another question you might want to consider.
For full functionality "Contact Form 7" actually requires three separate and distinct plug-ins:
- The contact form itself ("Contact Form 7")
- The CAPTCHA antispam feature ("Really Simple CAPTCHA")
- SMTP authentication ("WP Mail SMTP")
Search WordPress.org with those exact names for full details of each.
The CAPTCHA plug-in is by the same author as "Contact Form 7", which you might consider a put-off given his poor support for CF7.
Now, while I admit that all three do work well together, the inevitable question is: "why use three plug-ins when one would do?"
My philosophy in regards to these things is that I strive not to use, recommend or promote even a good product that has poor support. To my mind poor or non-existent support indicates contempt and/or disregard for the customer/user. And there's already too much of that on- and off-line.
I'd be very surprised if there isn't a single contact form plug-in that combines all the necessary features into one reliable package. So as time permits over the next week or so I'll be trying a few different contact form plug-ins to see if I can find one worthy of recommendation.
I hope this has been of assistance.