How can these people look at themselves in the mirror and claim that they are serving the best interests of their constituents?

It’s a complicated process, but in principle it works like this:

1. Convince yourself that there is a higher being which is benevolent, omnipotent, omniscient, transcendent and the creator of the universe, ignoring all logic and evidence to the contrary. It might be hard at first, but as your sanity starts to crumble away, it will get easier by the day – just hang in there, if Liquid Ocelot could do it, so can you!

2. Equip your fictional higher being with anthropomorphic features, sticking as close as possible to yourself as a template, so that your higher being will generally agree with your opinions and consider your personal lifestyle exemplary and worthy of promotion at the expense of all others. Important! Make particularly sure that your higher being hates all the people you dislike.

3. Get elected to office by fellow self-deluded religionists and a few dimwits who vote for you because they blame the other party for the consequences of your own party’s bad policies of the past.

4. Explain that you first priority is not serving men, but the god you created in your own image, because serving god is per definition in the best interest of your constituents. Congratulations! All of the effort to delude yourself is now paying off: you can freely give in to your worst, bigoted impulses while feeling upstanding and righteous – you can even feel victimized when anybody criticizes you for the ruined lives you leave in your wake!

Not to forget:

5. Profit!

How can these people look at themselves in the mirror and claim that they are serving the best interests of their constituents?

Well, if they don’t appear in mirrors and keep enough people under thrall, it would explain a lot.