Q:

Lorentz Transformation

I have specific questions about Lorentz transformations specifically about length contraction.

Why does length contraction only occur in the direction of travel, (not in all directions) when approaching the speed of light?
From an outside observer's perspective watching an object traveling close to the speed of  light, why does the obsever notice the object contract?

These Ideas are accepted by physicists, but for 1 I don't understand how they know that length contraction is in only one direction without they themselves being in the reference frame of someone or something traveling close to the speed of light.
For 2 Let's say that I'm driving in a car 5 meters long and I'm driving on a road that has white markers on it. Where the markers are in one line and each marker is 5 meters apart from the next one and the line of markers is parallel to my direction of travel. At normal speeds I would observe that at a given moment one marker would line up with back of the car and the next marker would line up with the front of the car (the car occupies 1 marker space). So now if I were to travel at speed such that my Lorentz factor is 2. I see one of two cases. In the first case I don't observe any change in myself or my car and I observe a length contraction of 2. Meaning that I would see a change in my environment not in myself or my car. So I would now observe that at a given moment the car would take up the length of 2 marker spaces. Therefore an outside observer would then see the car expanded by a factor of 2 not contracted. In the second case I do notice a change in myself. I contract with the space so things look very wierd in my perspective, Everything has contracted by a factor of 2 in the direction of travel. So my arms, my legs, the carseat, the hood of the car, etc. has all shrunk by a factor of 2. However because the road has shrunk by the same amount as the car, I take up only one marker space. Therefore an outside observer would not see any change in the length of me or the car. So in both cases the car doesn't contract from an outside observer's perspective.

A:

1) Why does length contraction only occur in the direction of travel,
  (not in all directions) when approaching the speed of light?

There are a number of ways to answer that vary in the level of sophistication.
But first, do understand that the Lorentz transformation gives length contraction only in the direction of motion.
So, if you accept the Lorentz transformation as the correct coordinate transformation between relatively moving reference frames, you accept that there is no transverse contraction.
Also remember, the Lorentz transformation was originally derived so that the measured speed of light would always be c in any inertial reference frame.
But, assume for the sake of argument that transverse length contraction does occur.  We almost immediately arrive at a contradiction.
Consider a wall upon which two parallel horizontal stripes have been painted.  The vertical distance between the stripes is exactly 1 meter which can be confirmed by placing an ideal meter stick vertically on the wall and finding that the ends of the meter stick touch the top and bottom stripes.
Now, imagine that the same vertical meter stick moves horizontally with respect to the wall.
If there is vertical contraction then, according to someone at rest with respect to the wall, the meter stick no longer reaches between both lines.  Since the stick is vertically contracted, the spacing between the lines is greater than the length of the meter stick.
However, to someone at rest with respect to the meter stick, it is the wall that is moving and it is the wall that is vertically contracted.  To this person, the meter stick overlaps the two lines; the spacing between the lines is less than the length of the meter stick.
But this is a contradiction!  It cannot be the case that the top end of the meter stick is both below and above the top stripe and similarly for the bottom end.
We conclude that transverse contraction leads to a paradox.

Therefore an outside observer would then see the car expanded by a
  factor of 2 not contracted.

No, that's not the correct conclusion.
First, the phrase "outside observer" is ambiguous.  What we should specify here is whether the observer is at rest with respect to the road or with respect to the car.
For an observer at rest with respect to the road, your car is contracted and fits in between the markers.
For an observer at rest with respect to your car, the distance between the markers is contracted and your car is longer than the markers.
This is not a contradiction because, and this is crucial, of the relativity of simultaneity; relatively moving observers do not agree on whether spatially separated events, along the axis of motion, are simultaneous.
In order to tell if the car fits between the markers, we must determine the location of the rear and front of the car at the same time according to spatially separated clocks synchronized in our reference frame.
So, imagine that there is a clock at the front of the car and at the rear of the car and that they are synchronized by Einstein synchronization according to the observer at rest with respect to the car.
Then, at some time, the location of the clocks are recorded.  According to the observer at rest with respect to the car, the recording of the location of the two clocks occurred at the same time.  Thus, the difference in the location of the clocks is the length of the car.
However, according to the observer at rest with respect to the road, the locations were not recorded at the same time, i.e, the clocks on the car are not synchronized.  Thus, the two observers do not agree on the length of the car.

