Trying to appear as if they are calling for justice
On the surface, most of these attempts take the form of a call for overdue justice. For example, when it comes out that Donald Trump shared classified information with the Russian Ambassador and someone demands that he be held accountable, the inevitable response is "but Hillary Clinton was never held accountable for her private email server!" The implication here is that whatever Donald Trump has done, it is a far greater miscarriage of justice that Hillary's email indiscretions went unpunished. An occasional variant of this goes "We know Hillary did something improper with having the private email server whereas the allegations against Trump haven't been proven, yet you're all concerned with nailing Trump while giving Hillary a free pass."
Accusing liberals of hypocrisy
This usually takes the form of "Look at how the media / Democrats / the person I'm talking to is all over Donald Trump but gave a free pass to Hillary / Obama / whoever over the similar thing they did. What a bunch of hypocrites!"
To obfuscate and distract
Most of the Trump supporters I talk to aren't dumb, so I think they grasp the incoherence of their approach on some level. However, just bringing up Hillary has the effect of changing the subject to something other than the misdeeds of the person they support. As Trump himself has demonstrated, obfuscation and distraction are effective weapons, and Trump supporters like nothing more than to rail against their favorite liberal targets anyway. This tactic is also especially effective against well-meaning interlocutors who assume the Trump supporter is approaching the conversation with the same good faith as they are and attempt to answer the Trumpy's charges against Hillary point-by-point, allowing the Trump supporter to succeed in changing the subject.
To garner sympathy
The Trump supporters often fall into this mode among themselves, but sometimes it bleeds out into discussion with people who don't share their viewpoint. The refrain goes like this "The media / Democrats / whoever are out to get Donald Trump. They never went after Hillary like this. The deck is totally stacked against him. See, unless you're part of the liberal establishment, you can't get a fair shake. Woe is us!"