At the outset, they supported Jeb!, then drifted to Rubio as Jeb flamed out. Finally to Cruz and/or Kasich. All the while, they hated Trump. They called him an incompetent, a moron, an imbecile. My aunt promised me point-blank "I will never vote for that idiot."
And then, at some point around the Convention, my aunt began posting soft pro-Trump messages—mostly in the form of digs against Hillary. These were mixed in with paradoxical articles with titles like "A Christian Case for Trump." (They're evangelical.)
When I confronted her and my uncle about her broken promise, she said "we're not voting for Trump, we're voting against Hillary." I think this is how many Republicans are justifying their awful choice. They'll never admit that they'd voted for such an historically terrible candidate, but will always frame it as voting against something. This logic completely frees them of any moral responsibility for Trump, and they can argue that they never supported his horrifying behavior.
Yet in order for this logic to make sense, Hillary has to be worse than Trump. So my aunt, normally a reasonable person, has bought into an increasingly perverse version of who Hillary is. Only a year ago, she told me how much she respected Hillary, despite their political differences. Now, Hillary's supposed crimes define her. And, by the logic, the crimes must be true, otherwise why would my aunt have voted for Trump?
I realize this is only one small case, but I think it may answer your question about how Republicans live with themselves.