How to approach Professors from other institutes (without potentially offending the ones at mine)
Most important, I question your statement that you need to "make it clear" that you are dissatisfied with your current faculty. That seems like a terrible idea unless you wish to break ties with your current institution. 

You can, of course, reach out to people about topics of interest without giving motives beyond your interest in their work and the answers they might give you.

But there may be little incentive for people to give you very much support, as they are busy and have their own responsibilities. But I don't think you improve that incentive, at all, by complaining about your current situation. 

If your emails are subject/topic based then word getting back to your current professors should have no consequences. But if you say offensive things, you can expect that people will take offense. 

But you might also ask yourself whether you are at the right institution or whether you should be looking for another.