Even if some people don't vote, wouldn't the interests of the nation be properly (statistically) represented by the people who do get out and vote?
This might be true if the actual voters are a representative sample of the population as a whole. But in practice, this is not generally the case. In the US, voting demographics are typically quite skewed in many ways.
- White people vote more than African-Americans and Hispanics
- Older people vote more than younger people
- The more education you received, the more likely you are to vote
Source: United States Elections Project
These demographic skews are correlated with party affiliation: the older, white demographic tends to be Republican, while the highly educated demographic favors Democrats.
The goal of these "get out the vote" campaigns is to increase turnout among the lower-represented groups. Although the postcard doesn't identify party affiliation, Democrats are most likely behind it, because polls have shown that a majority of Americans favor their policies (e.g. 55% like Obamacare and 75% wish to keep abortion legal), but voter turnout has typically been skewed towards Republicans. And Republican strategy has been to reduce turnout, because it favors them; see Is there evidence that the Republican Party leadership wants fewer people to vote?.