Why inverse of time delay is non causal
"Causal" in this case means "A leads to B". The cause is first, then comes the effect.

Example of causality: You hit your finger with a hammer, then it hurts.

Example of non-causality: Your finger hurts now, then 5 minutes later it gets hit by a hammer.

What it means in case of a signal delay (in DSP context, but you can apply the analogy to pure math): a delay block basically does this operation: "give me a sample number N from the past". It just seems logical - you can easily get data form the past - just place old samples in a buffer. The current state of the buffer (effect) depends on the previous samples (cause).

So what can be the inverse of a delay? "Give me a sample from the future". The effect has to depend on something that has not yet happened (the future), so this situation is called non-causal.