我明天开始放寒假。What is the meaning of 放？
The basic meaning of 放 is 'to release' 


放假 means 'be released (from job or school) for day off, vacation or holiday" 
放寒假 means 'be released (from school) for Winter holiday' 


Working men don't have Winter holiday, they have "新年假期/ 年假" (new year holiday)


放大假 = be released (from a job) for vacation 


Similarly:

放工 = be released (from a work day) = finish a work day and be let go = 'quit time'

放学 =  be released (from a school day) = finish a school day and be let go  = 'school out'

放监 =  be released (from jail) = finish jail sentence and be let free 


  开始 means 'start', so what does 放 mean here? 


As stated above '放' doesn't mean 'start'. If you are in the middle of Winter Holiday, you are "正在放寒假" 

开始(放)寒假  means 'start (being released from school for) Winter holiday' = 'start (having) Winter holiday'