Please review this answer first: Einstein's postulates ↔ Minkowski space for a Layman, and understand space-time pictures. Then you get "time dilation" and "Lorentz contraction" from two simple pictures.

Time Dilation

Time dilation refers a line which is in the direction of the time axis, but tilted a little to the right, and which represents the trajectory of the moving observer. This line has equally spaced dots along it's length, which are the ticks of the moving observer's clock. What is the vertical spacing of these equally spaced dots?

In geometry, they would occur with vertical spacing reduced by a factor of $1\over\sqrt{1+v^2}$. In relativity, they occur with vertical spacing increased by a factor of $1\over\sqrt{1-v^2}$ (in units where $c=1$). It's the same argument, but for the minus sign in the pythagorean theorem.

Length contraction

Length contraction refers to two parallel lines which are both exactly vertical. These are the two ends of a stationary ruler, and their length is measured perpendicularly between them to be $L$.

Now if you are a moving observer, your $t$-axis is tilted by a slope of $v$ relative to the original $t$ axis, and your $x$-axis is also tilted by a slope of $v$ relative to the original $x$ axis. So the actual length of the segment of your $x$-axis between the two ends of the rulers is the hypetenuse of a right triangle with sides $L$ and $Lv$. In geometry, it would be longer by a factor of $\sqrt{1+v^2}$, but in relativity, it's length is $L\sqrt{1-v^2}$.

Time contraction

If you turn the Length contraction picture on its side, so that the $x$-axis becomes the $t$ axis and the $t$-axis becomes the (negative of the) $x$ axis, then you get a strange picture of horizontal lines. These represent a line of simultaneous flag-wavers. They are everywhere in space, and they lift a flag up once a second.

If you move through these flag wavers in a rocket, and you look out the window to see how often the flag-wavers seem to wave to you, you see a different flag-waver wave a flag each time you see a flag waved. How often do the waves come?

The answer is that they come more frequently by a factor of $\sqrt{1-v^2}$. This is "time contraction", it is the length contraction picture tipped over in time.

Length dilation

If you tip the time dilation picture by $90^{\circ}$ into space, it gives you a simultaneity line for a moving observer. This line is marked up by flag-wavers at 1m who wave their flag exactly once.

If you have a moving system of flag wavers, and they all measure the distance between them to be $1m$, and they wave their flags once at exactly the same time as measured by them, how far apart are the flag-waving events as measured by you?

Because it is just a tipped over time-dilation picture, the answer is it is longer by $1\over\sqrt{1-v^2}$, the same time-dilation factor for a spatial interval. You can't get any of this straight without a picture, and it is just as obvious as Euclid's geometry, except you have to get used to minus signs in the pythagorean theorem.