This is easily derived from Pythagoras and a small thought experiment.
Imagine you have a setup of two horizontal mirrors facing each other. A photon is bouncing in between them at velocity c. The distance between the plates is d. The time it takes for the photon to travel from the top mirror to the bottom is t, which is equal to d/c when the frame of reference is stationary relative to the mirrors.
Now imagine that you are stationary and the mirrors are in a space ship travelling past at velocity v.When you look at the mirrors you will see the photon travelling a zig zag pattern through space. Draw this if you like to get an idea of what's happening. If you put a vertical line down from where the zig zag hits the top mirror to the bottom mirror, you will see a right angled triangle. The hypotenuse will be called d', the base -vd/c (from v = d/t) and the vertical side is simply d. Solving for d' is just a^2 = b^2 + c^2, and so d'^2 = d^2 - (vd/c)^2. This simplifies to d' = d*(1-v^2/c^2)^0.5.
Because d = c*t, and c is constant, d'/d = t'/t (where t is the dilated time).
So t' = t*(1-v^2/c^2)^0.5
A real life example is cosmic muons entering earth's atmosphere. They have a mean lifetime of 2.2 microseconds. At the speed they travel, without taking into account relativity, they should not reach the earth's surface before disappearing. However, because of the relativistic time dilation/length contraction, they do reach the earth, as the distance is shortened by (1-v^2/c^2)^0.5, or alternatively, they have more time as time has slowed down for them by the same factor. Time dilation and length contraction are the same thing, just focusing on different aspects of space-time.
So from Tim's perspective, using the example figures you gave, he has travelled 0.1 ly in 0.1 year. Space was contracted, and because he was travelling close to the speed of light, it only took him 0.1 years. Alternatively, the time it took him to travel became shorter, and as he was travelling close to c, he travelled a shorter distance.
From Jim's perspective, he has travelled 1 ly in 1 year.
As for the 'paradox' part of the twin's paradox, it isn't a paradox because it doesn't count in accelerating reference frames.