"Speculative Execution" is gonna be my boat name.

I'll tell people it's a punny triple entendre, leading people to believe I am a wall street hacking guru hit-man.

I know what speculative execution is and have some understanding of spectre already. What I don't get is what it means to exploit a speculative race condition.

So the thing with speculative attacks is that you can use an invariant that is broken speculatively to leak things. In this case they use a speculative race condition to gain speculative code execution, much like you can use a normal race condition to gain real code execution. Arbitrary speculative code execution can be used to leak data which can be picked up via side channels.