Why my boost converter is not boosting the voltage?
After a quick look I guess because you didn't follow the typical applications of the datasheet. The circuit you are showing here suggest you want to use the LT3517 as "Boost 100mA LED Driver with LED Open Protection".
And that means you are missing the feedback resistor network R1/R2 on the FB pin.


Try to add these or if you didn't add them on purpose tell us a bit more detail on what you are trying to do.

edit
Looking at your comments and question again I realized that you might have another understanding problem.
You are showing the VLed graph and claim that the voltage is not getting boosted. What you see here is the led voltage drop for the current you set with the current sense resistor. Check the current going through the Led and you should see a constant current through the Led that is getting regulated by the IT.