T-SQL | How to SET Value dynamically to Variables in While Loop

I think you don't need to use variable here :

INSERT INTO VarTable (Var1, Var2, Var3, Var4, Var5, Var6, Var7, Var8, Var9, Var10, Var11, Var12)
      SELECT SUM(CASE WHEN MonthCol = 1 THEN 1 ELSE 0 END), 
             SUM(CASE WHEN MonthCol = 2 THEN 1 ELSE 0 END),
              . . .
             SUM(CASE WHEN MonthCol = 12 THEN 1 ELSE 0 END)
      FROM #TempTable t;

However, here conditional aggregation is enough as there is only 12 months. So, dynamic approach is not needed.

well this is what you can try here:

Declare a table variable with 2 columns

 declare @mytable table(
   VariableName varchar(255),
   VariableValue int
 )

Now Declare 2 more variables, one to set value and second to iterate through loop

 Declare @value int;
 Declare @month int = 1;

Now do the while loop like this:

 while (@month <= 12 ) --since there are only 12 months
 begin
   set @value = (SELECT COUNT(*) FROM #TempTable WHERE MonthCol = cast(@month as varchar));
  insert into  @mytable values ('Month'+cast (@month as varchar), value );
  set @month = @month +1;
 end

Now you'll have your DataSet in table form which you can access by using the simple query:

VariableName   VariableValue   
-----------    -------------- 
   Month1          23
   Month2          19
   Month3          66
   Month4          12
   Month5          05
   Month6          04
   Month7          63    
   Month8          76    
   Month9          21    
   Month10         23    
   Month11         63    
   Month12         12  



Select * from @mytable where VariableName = 'Month1'