there are many ways to achieve what you want. Basically you want to remove is and this 
so either you can do this:
a. If words you want to remove are always the first and the second one then
 var filteredArray = str.split(" ").splice(2);
b. If words you want to remove can be anywhere in the string
var  filteredArray = str.replace(/this|is/g, function(w){
        switch(w){
           case 'this':
                 return '' ;
              case 'is':
                   return '';
          }
    }).trim().split(' ');
c. There is one more lame way to do this.
var str= "This is Javascript String Split Function example ";
var parts = str.split(' ');
var clean = [];
for (var i = 0; i < parts.length; i++) {
   var part = parts[i];
   if (part !== "This" && part !=="is")
      clean.push(part);
}
d. using  simple regex to replace multiple words
var str= "This is Javascript String Split Function example ";
   var words = ['This', 'is'];
   var text = str.replace(new RegExp('(' + words.join('|') + ')', 'g'), '').split(' '); 
based on your need you can go for any one of them.
references : this , this and this