Here's a thorough analysis

The problem:

1. Kvothe is cold, he doesn't have a way to get warm, and is in danger of going into hypothermic shock.
2. There are still a lot of enemy soldiers that will probably kill him before the hypothermia actually kills him.

Notes:

Most of the sympathetic links mentioned in TKC are related to "energy money changing", however electrical and chemical properties are also subject to sympathetic links.

For example: when Abenthy asks Kvothe how he would bring down a bird in flight, he has the idea to bind a similar feather to the bird's feather with a 2nd catalytic binding and using soap to dissolve the oils that make the bird's feathers smooth (unfortunately he doesn't have a feather, so he has to think of something else).

We also know from some of his interview questions that there are bindings having to do with electrical (galvanic) properties, and he also knows a great deal about electricity from his work artificing, also evidenced by interview questions, e.g. galvanic properties, reference to capacitance etc.

Lightning is an electrostatic discharge, and in order for a discharge to happen there needs to be a high potential difference between two regions of space. The further those regions of space are from one another, the greater potential difference is needed.

We know from stories told at the university that when a sypmathist moves energy around that some portion of it goes into their own body, with often dangerous or deadly consequences, usually in the form of heat.

Kvothe's idea and "solution":

1. Bind two similar arrows together using a galvanic (electrical) binding
2. Bind the arrow being shot at the tree with the tree, the link becoming much better when the arrow is embedded into it. (it doesn't state this binding in the book, but he does mention 6 separate bindings and this is a likely candidate)
3. Bind the arrow he keeps to the ground, also becoming a better link when he physically plunges the arrow into the ground at the dramatic moment. (also not explicitly stated)
4. Duplicate some or each of these bindings to make them stronger, which is the reasonable assumption because he is making six bindings with the six parts of his mind. It's probably not just 2 each, my guess is 2 to the tree being a medium good link, 1 between the arrows being a great link, 3 between the arrow and the ground being a pretty bad link (wood to dirt/mud).
5. These bindings cause the potential difference between the sky and the tree to be the same as it is between the tree and the ground, but the tree is much closer to the sky. This is consistent with Kvothe's explanation that it was "An attempt to ground the tree more strongly than any lightning rod." This causes the electrostatic discharge (lightning) to strike the tree multiple times.
6. Some of the energy traveling through the link ends up going into the body of the sympathist, which we know from conversations with Will and Manet, this warms Kvothe's body and he's lucky enough to gain enough heat to avoid hypothermia and not so much that he cooks his own flesh. He then looses consciousness and the links are broken.

Whether other forces were at work to ensure the success of this strategy is a matter of some debate, but we don't have any evidence of this other than that Kvothe was exceedingly lucky to come out of that situation alive.

He made a lightning rod through Sympathy... with some amount of luck

Kvothe very clearly prepares to use sympathy as he divides his mind in six parts before binding the arrows. Later he explains the nature of the bind, without discarding some luck coming into play.

The lightning? Well, the lightning is difficult to explain. A storm overhead. A galvanic binding with two similar arrows. An attempt to ground the tree more strongly than any lightning rod. Honestly, I don't know if I can take credit for the lightning striking when and where it did.

Rothfuss, Patrick. The Wise Man's Fear: The Kingkiller Chronicle: Day Two (p. 619). Penguin Group. Kindle Edition.