I don't understand from the very first step of how the equations came to be. Is there a different way to solve this? So I can understand it intuitively instead of algebraically?

This solution is not hard to understand. You're breaking the problem up into two parts. The first part is when the car moves from tail to head, which takes a time "t." If "r" is the walking speed of the dragon, then 3r is the driving speed of the car (this is given in the problem). In time t, the car moves a distance of speed*time = 3rt. HOWEVER, the distance travelled by the car must be equal to the length of the dragon PLUS the distance travelled by the dragon in time t, which is rt. Hence:

100+rt = 3rt

This equation is saying "distance travelled by dragon + length of dragon = distance travelled by car"

Now consider the second part, when the car moves from the head back to the tail, which takes an amount of time "y". In this case, the distance travelled by the car is the length of the dragon MINUS the distance travelled by the dragon. It's minus, because the car has to go less than 100 ft to reach the tail, because the tail is moving towards it at speed r.

The distance travelled by the car is: 3ry
The distance travelled by the dragon is ry