The speed of light is a result of the laws of electromagnetism. No matter how fast you move the laws of electromagnetism and therefore the speed of light wont change.

I know of no way to make it intuitive though. Things improve slightly if you replace speed with 'rapidity' which places the speed of light at infinity, but then you get that objects with infinite rapidity don't travel instantly from one place to another (although this is true in some frames of reference).