Here's the part of the text that is the source of that "embarrassed":
Anxieties run higher among Republicans in large part because of the ferocious and at times juvenile nature of the insult-laden campaign, which has featured taunts over character and even manhood as much as serious policy debates. About six in 10 Republican primary voters say the overall tone of their party’s nomination fight has been more negative than in past campaigns, while only one in 10 Democratic primary voters hold the same view of their party’s campaign.I went into the PDF of the poll to get the "embarrassment" question. It's: "So far, do you think the Republican presidential campaign has made you feel MOSTLY proud of the Republican Party or MOSTLY embarrassed by the Republican Party?"
And 60 percent of Republican primary voters said the campaign had made them feel mostly embarrassed about their party, while only 13 percent of Democratic primary voters expressed that opinion.
So "embarrassed" is the pollster's word and your only options were "embarrassed" or "proud." Personally, I wouldn't know how to answer that question. I don't have enough of a feeling of identity with a party to feel embarrassed or proud of it. It's not me. It's not my child.
But in any case, you can't figure out from the answer what is embarrassing the people who are embarrassed. The NYT — in writing "ferocious... juvenile nature... insult-laden... taunts" — seems to assume it's Trump's manner of speaking. But there are other things one might be embarrassed about — the way the elite insiders are disrespecting the people Trump is energizing, the failure of low-polling candidates to withdraw in time to let those with a chance build up sufficient support to balance Trump's fast start, or maybe just the way the media is continually advising them that they ought to be embarrassed.