to answer this question .... i'll have to again borrow someone else's work, this time brian greene's ... i guess u could just get urself a copy of the elegant universe, and he explains ur exact ur question on P38-40 ... if u can bear with me then i'll try my hardest to explain ...
imagine a simplified light clock that consists of two parallel mirrors with a photon that bounces between them. the clock will tick each time the photon completes a roud-strip journey
now we have another clock which moves at a constant speed:
(pic1) (pic 2) (pic3) (pic4)
------- ------- ------- -------
* * * *
------- -------- ------- --------
these 4 pix represent the SAME CLOCK but at diff time intervals .....
so looking at these 4 pix, we know that the photon is NOT moving just moving up and down, its mvoing side ways at an angle as well ...
------ ----- ------
. * .
------ ----- -----
(the fullstops represent the diagonal paths the photon is taking as the clock moves across the surface) while the stationary clcok will only allow its photon to move up and down:
now common sense tells us that the diagonal distance will always be GREATER than the vertical one. the speed of light canNOT be changed, so we can derive that the moving clock will tick LESS frequently, since it must travel FURTHER to complete the one tick jouney. this in turn implies that the moving clock will tick slower than the stationary clock.
i hope the above has made sense to u, if theres any bits u dont understand pls inform me about it :o)
or alternatively as i have already suggested, pick up the elegant universe ... iv neva acutally finished reading it, iv been stuck around half way thru the bk coz i dont hav the mental capacity to keep going lol (too diff for my brain cells to digest) but the first couple of chapters on relativity is EXCELLENT!!! very detailed and comprehensible! :o)
hope iv helped u out and answered ur question :o)