Christians are often asked what right we have to judge others. This is based on a fundamental misunderstanding of what it means to judge. You can judge two things: The inward heart, and the outward appearance. The inward heart, and the end destination of a soul, is for God and God alone to judge. The outward actions, however, are a thing we can all make righteous judgements about.
Think of it this way: Who has the legal authority to throw a thief in jail? Not me. If I catch a thief and detain him, that's called kidnapping. But can a criminal court convict that same thief? Yes. What can I do? Well, if I see that thief stealing, I can say to him "You're breaking the law, you should return the property you stole immediately". Is it logical for the thief to then say "who are you to judge me"? And what if his friend then chimed in "yeah, who are you to judge him? It's not against the law to steal. You're just a hateful bigot"?
As Christians, it is entirely permissible to compare the law of God to the outward actions of our fellow man, just as it is entirely permissible for a citizen to compare the law of the land to the outward actions of their fellow citizens. If you break the law, I can tell you. If you sin, it would actually be very cruel of me not to tell you.
See, first of all, the Bible is flooded with calls to Christians to preach. It is an inherently Christian thing to do. If you're a Christian, but you're not preaching, you're doing something wrong. But more importantly, while I, myself, will not be dealing out the consequences of your actions, these consequences do exist.
This is a very important thing. Just as a Christian is not able to send you to Hell, we are also not able to send you to Heaven. As a matter of fact, it is purely by grace of God that we, ourselves, get into Heaven. We're not going because we deserve it; we don't. We're not going because we're entitled to it; we aren't. Every Christian on this planet deserves to go to Hell, because we've all done things that violate God's Holy law. However, He loves us enough to grant us grace. And we want to share that grace.
As Christians, we have zero power over Heaven or Hell. What that means is that, ultimately, all this whining at Christians is silly. There's no point calling us bigots, or judgemental, or hateful, or >insert random phobia here<, because at the end of the day, it is no more our right to justify your sin than it is to judge you. It won't be us you face on judgement day, but God. He will judge you according to your faith and your works, and it is His absolute right to do so, because everything is His. Even your body does not actually belong to you, but to Him.
So what can Christians do but preach? It would be nothing short of cruelty to fail. We believe, as we have been told by God, that all have sinned, and fallen short of His glory, but by His grace, He has provided a way to avoid our just punishment. He sent His Son to become sin, though knowing no sin Himself, and die on the cross. After that, He rose again, so that all who confess Him as Lord, and believe He rose from the dead, will be saved.