Oof I've been there. Not a tutor but when explaining concepts to my friends, especially ones which rely on intuition, it's hard because everyone doesn't have that level of understanding. Personally, I resort to questions here. Instead of outright explaining stuff, try asking your students questions. Like, if you need to explain the concept of derivatives, instead of teaching the first principle or deriving the derivative directly, talk to them about change, relate it to common notions of change like speed (which is my personal favorite example for explaining integration and differentiation), and then eventually veer them into the actual subject, all while asking questions so that *they* arrive at the answer themselves. Of course, help them out if they're extremely stuck in some place. I find that when people can relate a Math concept to something they are more familiar with, they understand the concept in a much better way, especially when they arrive at the solution themselves. It's a weird, euphoria like feeling.
>Also another question I just thought about, imagine a kid that just doesn't get addition, no matter how I approach it, how can I help him understand? It seems to me so fundamental but I imagine many struggle with it.
So, here's what I'd probably do. Give that kid an object (maybe a candy, because why not lol!). Then, give him another. Then, give him one more. Ask him what you're doing. He'd probably say you're giving him more and moe candies. Then, while giving him more and more candies, ask him if he's able to hold onto them. He'll probably say no after some time. Then, tell him that you were "adding" candies to his hand, and relates this to the fact that he was having trouble handling "more" candies. So, the fundamental concepts I'm taking about here is that "addition is the process of adding more of an object to an already existing amount of an object" (Don't say this part to them lol! This is just for your reference)
Hope this helped?
Edit: As for the trig part, maybe handle it like this. Give the students a right angled triangle. Give them only one side and one angle. Now, ask them to find the other sides. Obviously, without trig, it wouldn't be possible (unless the triangle is a special one, where one would know the answer right away). So, when they arrive at the conclusion that this problem cannot be solved with basic arithmetic and geometry, introduce the idea of trigonometry, which literally stands for study of the measure of three sides. So, this is another way to teach a new concept: "Enforce their existing ideas and show them that a new concept is needed to solve a higher problem". Once they get the basics of trig, explaining identities as derivations is probably the best way to go about it!