TM to detect repeats of any other TM and input value
I think you're wondering how one Turing machine can keep track of what another Turing machine is doing, including checking that machine's state and knowing what's on its tape. After all, Turing machines don't have a method for "outputting" a description of what they're doing&mdash;they merely run.

But a Turing machine that takes in a description of another Turing machine can simulate that second machine on its tape, the same way that you can simulate running a TM using pencil and paper. The TM writes down the current configuration of the simulation, including what the simulated current state is, what the second machine's tape says, and where the machine's tape head is. It follows rules for moving the tape head, writing to the simulated tape, and changing the simulated current state.

That's how one machine can simulate another. Another helpful trick is that a TM can have any finite number of tapes without affecting its capabilities&mdash;a three-tape TM is no more powerful than a one-tape TM. So if it's conceptually helpful, you can give the TM a separate tape for each concurrent simulation.