Unfortunately, most students haven't been taught mathematics well at any point in their lives. The week before the fall semester starts I teach an algebra review class to incoming freshmen, and I'm amazed at how poor some of their math skills are (for example, confused why 1/x + 1/y is not 1/(x + y), or reducing (5 + x) / (5 + y) to x / y).
It's a travesty that students can get this far in life without really understanding what's going on, but I don't think it's all their fault either.
Fortunately, they have you! Better late than never, as they say. So here's my advice on teaching mathematical rigor to people:
1. Be as excited as you can be. Rigorous argument is not necessarily the most enthralling of things, but people pick up on whether you care about something and that can make a big difference in their opinion of the topic.
2. Be patient. You've, perhaps subconsciously, spent years developing the understanding you have of the subject; they have not. That can change, but it won't happen right away.
3. Ask them questions. Building connections between ideas might come naturally to you, but it does not to everyone. Try to nudge them to see relationships between things, even if you/they don't fully explore that relationship and why it exists right away. Walk them through your process of seeing why a solution is right or wrong.
4. Think about what guides your intuition for problems and explain what you can of that process.