One reason for the difference is time dilation--a given coordinate clock at rest in one frame will be running slow as measured by coordinate clocks at rest in another frame. It can be demonstrated that this follows logically from the two postulates of special relativity, using the "light-clock" thought-experiment detailed here. The second postulate says that both frames must measure light to move at the same same speed of c, so the thought-experiment uses a hypothetical type of "clock" which measures time by light bouncing back and forth between two mirrors, and if both frames measure the light to move at c this implies that the frame in which the clock is moving will measure it to tick slower than the frame where it's at rest. And the first postulate says all laws of physics work the same way in all frames, so if some other type of clock would keep the same time as a light clock if both were at rest in some frame (say, the frame of a lab on Earth), this must be true for a light-clock/other-clock pair in any frame, thus all types of clocks must display the same time slowdown as a light clock when they are in motion relative to the observer.
The other reason for the difference in coordinate time between frames is the relativity of simultaneity, which has to do with the rule that SR uses for how different coordinate clocks in the same frame should be "synchronized", namely the Einstein synchronization convention. This convention says that if I have two coordinate clocks at rest in my frame at different locations, I should synchronize them using the assumption that light signals travel at the same speed in all directions in my frame (the second postulate again). So if a light signal is sent from clock A when it reads time T1, and it's reflected by a mirror next to clock B when that clock reads T, and it returns to clock A when it reads T2, then A and B are defined as "synchronized" if T is exactly halfway between T1 and T2. Another equivalent method would be to set off a flash of light at a point midway between A and B, and then make sure that they both read the same time when the light from the flash reaches them.
But using this synchronization method, it's not hard to deduce that a pair of clocks which are synchronized in their own rest frame can be out-of-sync in the coordinates of other frames, given that an observer in another frame assumes that light travels at the same speed in all directions relative to herself. For example, suppose Bob is aboard a train which is moving forward relative to Alice, and he decides to synchronize clocks at the front and back of his ship by setting off a flash midway between them and setting them to the same time when the light reaches them. From Alice's perspective, the clock at the back of the train is moving towards the point on the rails that was next to the flash when it was set off, and the clock at the front is moving away from that point on the rails, so if she assumes both light rays move at the same speed away from that point, light from the flash must reach the back clock before it reaches the front clock. Thus if Bob sets both clocks to read the same time when the light hits them, in Alice's frame the back clock will end up having a time that's ahead of the front clock's time.
