Why 给 is used in 我的手被虫子给咬了? I saw this in the subtitles of a TV show:


  我的手被虫子给咬了。


Why 给 is used in that sentence? 

Without considering 给, I've translated it to:


  My hand was bitten by an insect.
给 is part of the passive structure of the sentence you've cited. It can be used before the verb either with or without another passive marker (like the neutral 被 or the colloquial 让 or 叫). It doesn't change the meaning of the sentence though, as dacongy mentioned, it can add emphasis.
Here are some example sentences:

衣服［让］他给弄脏了。
The clothes were dirtied by him.
这个杯子［叫］我给打碎了。
The glass was broken by me.

Confusingly, 给, like 叫 and 让, can also be used as a passive marker by itself in sentences like the following

门给风吹开了。
The door was blown open by the wind.
孩子给妈妈打了一顿。
The children were given a beating by their mother.

In the above sentences, 给 could be replaced with 叫, 让, or 被 with no change in meaning. (The register and regional flavor will be slightly different.)
This paper, &quot;北京话“给”字被动句的地位及其历史发展 (PDF)&quot;, which I haven't yet had a chance to read, analyzes the use of 给 in passive sentences (被动句) in Beijing dialect. I skimmed the abstract and it seemed like it might be illuminating.