You say "RDR is showing reading garbage". But you are not accessing RDR directly. Do you mean your rx buffer shows something other that what you expect? If so, what were you expecting to receive and what did you actually receive? When you first transmit your canned "aTxBuffer", does whoever receives that get THAT data OK? Can you loop-back your TX to your RX (perhaps on the far side of the LTC2873) and see if you can receive the same thing that you send.
I don't see you supplying an error callback function. Do you know if any of the error bits are getting set?
You configure the LTC control lines then immediately start sending. Does the LTC part need any specific delay between changing control lines and actually being able to rx/tx data?
Are you currently looping back the serial port, such that you are expecting to receive the same 5 bytes that you first sent before the while() loop? If so, the 1 second delay after the HAL_UART_Transmit_IT() call will cause you to miss that data.
FYI - it is perhaps better forums etiquette to actually include your source in your post as opposed to requiring us to download it. As your user name implies, I am also a control freak and very hesitant/paranoid about downloading files of unknown content, even from forums that I hope and presume would filter files before allowing them to be posted :)