I'm having trouble understanding Lorentz transforms

[A.T.]

I don't see how relativity of simultaneity is important

Maybe this video will be useful:

 

[Fredrik]

But equation with t'a=t'b yields something of the form x'<x. While equation with ta=tb yields something of the form x<x'. Why isn't this contradictory?

In the case of time dilation, both observers will consider the other one's clock slow. This sounds like a contradiction until you realize that statements about an observer's point of view are really statements about numbers assigned by a coordinate system that we associate with his motion. Since the two observers move in different ways, we associate different coordinate systems with their motions. So there's nothing inherently contradictory about a disagreement between two observers describing the same thing in different ways. Look at an object that's now on your right. An observer at your location facing the opposite direction would say that the object is to the left. This is clearly not a contradiction.

Further, in the typical SR scenarios, the two observers (and their coordinate systems) aren't describing the same thing. In the case of time dilation, A is describing a segment of B's world line and B is describing a segment of A's world line. In the case of length contraction, they are describing different line segments (the red and blue lines in my diagram) in the rod's world sheet.