I am trying to achieve this look in illustrator.
I can only do this much using gradient tool in illustrator.
Can someone tell me how can I make it look fade out like the first picture?
Use Opacity Mask
It's lifesaver when you need to make something complex comps :D I personally use it a lot in both Illustrator and Photoshop!
I'll try to explain the steps which I personally Follows!
Make sure you have opened Transparency window by Shift+Ctrl+F10 or window->transparency.
Create a rectangle or any shape and fill a gradient color in it but make sure the both color should be pure black and pure white ( basically the white colored part will be shown and the black colored part will be hidden after mask )
Select rectangle and your path and click on Make mask :D and voila you're done.
I shared little video to help out more! Cheers!
This is somewhat guesswork since I can't see the mesh points in your screenshots.
However, I don't see why you need to mask anything.
Merely select the mesh point(s) on the left and set their color to white. They will fade the mesh to white on that side.
Sometimes with meshes it's easier to start with rudimentary shapes and adjust. This is what I would do here -- start with a simple rectangular mesh and then move and adjust the mesh points to match the desired shape.