Outgoing Mail Settings, CiviSMTP & PHP 5.6.x Problem
The recommended and least invasive fix is to resolve the underlying SSL issues. 

It was intentional that PHP5.6 tightened security here. If your hosting environment doesn't validate the SSL certificate you're communicating with, one of these things is probably true -


The hosting environment has a misconfigured certificate chain. (Fix varies by hosting environment - for Debian, apt-get install ca-certificates - there's probably already a Stack Exchange answer for any other specific OS/environment a site is hosted on.) 
The remote service requires that you install a custom CA chain before their certificate validates. (Perhaps your SMTP provider uses a self-signed SSL certificate, or a cert from a certificate authority not installed on your hosting environment.)
The remote service has a misconfigured/bad SSL certificate. (Ask remote service to fix SSL.)
The connection between the two is being interfered with. ( )


Identify whether it's your hosting environment or the remote site's SSL which is failing, by testing multiple remote sites from multiple client environments. This SE answer shows how to test with openssl command from CLI and has links to web services for same.

Then resolve the SSL issues at whichever end they are occurring, until SSL validates and your CiviCRM is able to connect again.

This assumes that you can resolve the underlying SSL issues! If the remote site offers SSL without a valid chain, or uses a certificate which your server can't make sense of (ex: this libressl bug - as opposed to a valid CA you just don't have installed), then you just can't have SSL validation. 

Alternative solutions (not recommended)

Unless you're running CiviCRM for local development and with fake contact data,


Don't disable PHP5.6's SSL validation by default. This is best avoided.
Don't downgrade to PHP5.5. This combines disabling SSL validation with downgrading to an unsupported version of PHP, so it's worse!
As of early 2017, OSX / CiviCRM / Gmail may also be affected by this homebrew issue or this libressl issue. (That's an unlikely combination to find in production though.)