To me the political definition of "connect" simply means recognizing which messages work and which don't. We've all seen candidates who "misread" the voter tea leaves, while others seem to have that knack of saying the right thing in the right way (in my mind Reagan and Clinton were among the best in that regard). Connect doesn't mean "having the same experiences as" voters.
Sort of the distinction between having empathy vs. sympathy. You want politicians to have sympathy for the plight of the common man without necessarily the empathy of having gone through it.
I personally have never understood the appeal of the populist "common man" national politician for two reasons 1) most of the time their "common man" status is a fabrication (as go deftly points out above) and 2) I want our national leaders to be exceptional, out on the far right tail of the bell curve, not in here near the mean with the rest of us schmucks (sidebar, I learned a few days ago that schmuck is Yiddish for penis ... didn't know that).
So in that vein, I don't really care whether a presidential candidate "gets" me. I hope he/she is far smarter, capable, and deft than I am. If he/she has never spent a day as an adult slogging it out in the middle class, that's fine, so long as they are able to sympathize with how their decisions do ultimately affect me (and others).