What is the meaning of 一看 in 打开盒子一看，里面是一只小猫?
打开盒子看，里面是一只小猫 (open the box and look, it is a kitten inside)
  


看 here is a simple verb 'to see'


  
  打开盒子看看，里面是一只小猫 (open the box and briefly (or casually) look, it is a kitten inside)
  


看看 here is a reduplication of verb ; reduplication verb imply the verb is occurring in a brief time period, in a slight degree or in a casual manner. 


  
  打开盒子看一看，里面是一只小猫 (open the box and looked once, it is a kitten inside)
  


看一看 (look one look = look one time = looked once) is  [verb + counting word + verb classifier]; the same structure can be applied to other verbs, e.g. 打一打 (hit one hit = hit one time = hit once); you can replace '一' with different counting words like '看幾看' (look few looks = look few times)


  
  打开盒子一看，里面是一只小猫 (open the box and quickly look, it is a kitten inside)
  


一看 here is [adverb (一) + verb (看)] ; '一' as an adverb means 'once, abruptly, quickly or briefly'

Example of '一' as an adverb:

他心中一怒 (not 他心中怒一怒) means "he abruptly get angry inside"

曹操一听大怒 (not 曹操听一听大怒) means "once Cao Cao heard it, he got very angry"

我一想不錯 (not 我想一想不錯) means "once I think about it, I know it is correct"

他终于能一展抱负 (not 他终于能展一展抱负) means "he finally can extend his ambition for once" or "he finally can extend his ambition abruptly" 

他一改风格 (not 他改一改风格) means 'He abruptly changes style' or 'He changes style for once')