What is Attach Reject in LTE?

It is the EPS NAS message the network sends to indicate that the corresponding attach request was rejected.

What should I inspect first in Attach Reject?

Start with the EMM cause, then inspect any returned T3402, T3346, or optional ESM information.

Why is Attach Reject important in troubleshooting?

Because it tells you not only that attach failed, but also which reject branch and retry behavior the UE is expected to follow.

What usually comes after Attach Reject?

That depends on the returned cause and timers: the UE may retry later, wait in backoff, or remain out of normal attach continuation.