I've read about special relativity recently and I've learned about the Lorentz transformation. I get what time dilation and length contraction are and I think I understand what the transformation does (at least to the distances and time intervals) but I want to be able to show it mathematically in a simple and intuitive form. I've read a lot of papers yet it is quite hard and unintuitive to me to derive the formula for time coordinate. I don't know if such a complex topic can be explained simply but what I'm looking for is a way to derive these equations not by squaring, substracting the sides etc. which can't be showed visually and doesn't represent any "ideas" in physics but by explaining which part of the formula means what.
Here's a link to a thread I've started on r/askphysics, maybe it'll be useful.
I appreciate all the help.