It's less about 'ashamed of their opinions' or the 'shy trump voter.' The article talks about this. The crux of the problem - at least argued by this authored and from what I see working politics backed up generally - is that those Republican / Trump voters do not trust others and are less likely to take the poll in the first place. Those that do take survey generally do not lie about their support.
The article gives evidence. Here is the section:
"I use the General Social Survey’s question, which is, “Generally speaking, would you say that most people can be trusted or that you can’t be too careful in dealing with people?” ....
It turns out, in the GSS, that 70 percent of people say that people can’t be trusted. And if you do phone surveys, and you weight, you will get that 50 percent of people say that people can be trusted. It’s a pretty massive gap.
[Sociologist] Robert Putnam actually did some research on this, but people who don’t trust people and don’t trust institutions are way less likely to answer phone surveys. Unsurprising! This has always been true. It just used to not matter."