Leaders and politicians will naturally claim that their policies are wise, their support is strong, and their popularity is high. In situations where public outcry or media coverage creates the strong impression that one or all of those is not true, then politicians may choose to resort to defensiveness and working the ref. Claiming that the silent majority supports you is multiple arguments and tactics rolled together.
First, it's a claim of popular legitimacy, a must for any leader but particularly one that claims to be representative.
Second, it appeals to some people who may not truly be silent but who feel embattled or outnumbered; it bolsters your supporters to make them feel like defenders of the true majority, rather than doomed dead-enders of a rump movement. This is particularly important for internal morale within a governing group in an effort to keep associated politicians and staff on your side rather than defecting or withdrawing to the sidelines.
Third, it pushes back on media and on opposition politicians, suggesting that they are biased and carrying water for a small group of radicals and failing in their various duties to cater to the broad majority. If media or opposition types are challenged, the hope is they may slow or soften their criticism to seem less outside the mainstream.
Variants of the Silent Majority argument are regularly used in various forms by pundits who feel embattled or undervalued. A recent permutation is the argument by pro-administration pundits that ACA/Obamacare would be more popular as people were familiar with it or, in an earlier permutation, as President Obama gave more speeches about it. In hindsight, neither of these have yet proved true. This is similar to the Silent Majority argument because it suggests that the town hall arguments and Tea Party protests were a non-representative sample of loud miscreants and malcontents, and that the true majority of people would appreciate the president's policies. That's more or less exactly the Silent Majority argument.
The previous administration relied on a modified Silent Majority argument with regard to Iraq, when President Bush, VP Cheney and staffers argued that the broad majority of Iraqis wanted US intervention to be liberated from Saddam. This argument was relatively plausible for a brief window in 2003, but quickly thereafter seemed to be more like wishful thinking. This is a modified version because it is not arguing that a majority of the constituency approves of a policy, but that a majority of the affected population on non-constituents approves.
Of course, the true draw of arguments in the Silent Majority style is that it is non-falsifiable so long as it is somewhat plausible. If the argument is that a majority likes your actions in a way that is non-apparent and possibly non-verifiable, then politicians can make the claim and nobody can truly disprove it. Even if half the country disagrees, if your political party mostly accepts the Silent Majority argument, then it's not a lie but something closer to an opinion or a judgment call. The best lie for a politician is one that nobody can catch you making.