Others have dealt with the language question, so I'll just add that backyard astronomers know that "a fist held at arm's length" subtends an angle of about 10 degrees. It is a common aid to "star-hopping," locating things in the sky visually by looking at the sky, using pairs of stars as pointers, and comparing the sky to a printed star atlas. That is just the angle (which could be measured in degrees or radians), which would be used to measure the "distance" between two stars. As Gary says, to know how big a piece of sky the fist covers, you would need to know the SOLID angle in steradians.

'At arm's length' refers to holding your arm out straight. If you do that, your fist (your clenched hand) covers a certain area. It is that area of sky that it is referring to, not the length of the arm. If you want a more technical answer, it is giving you a simple reference to a certain 立體角, or a number of 球面度 (steradians). I did a quick check, to try to find out exactly how many steradians a fist at arms length covers, but was surprised at how hard it was to find out, sorry!

It means that Kepler observed a section of the sky which is as long as your arm if you stretch it out so it was straight and not bent. So imagine pushing your fist away from your body as far as you can reach. :)

"One" refers to "piece of sky". So this paragraph could be rewritten as: During its four-year mission, Kepler observed a tiny, random, average piece of the sky. The length of the piece of the sky was as long as your arm if you were to reach your fist out from your body as far as you can. More than 1,000 planets have been discovered so far from just the nearby stars in that tiny patch of the sky.