(1) Concessionary means that someone is willing to cooperate, usually by giving up something (from the word "concede"). Tone means someone's attitude. So putting the two together, it means that the senator has a cooperative attitude - she is willing to work with other people.
(2) "To Strike a ____ Tone" is a set phrase in English, and it can be used to describe someone's tone. You can think of it as "To Have a ___ Tone" - in this case, it means "she has a concessionary tone". Refer to (1).
(3) Hardline means not willing to change or cooperate. Loath means that you don't want to do something. It means you are reluctant, unwilling. With this in mind let's look at the sentence as a whole.
Despite her hardline position on tax reform:
This means that she was not willing to change her position on tax reform (hardline). The word despite will make sense after seeing the second part of the sentence:
the senator was not loath to strike a concessionary tone when she debated the issue with her opponents.
The senator was not unwilling, or reluctant (loath) to cooperate (strike a concessionary tone) when she debated the issue ("the issue" refers to Tax Reform from the first part) with her opponents.
Now we see the purpose of the word "Despite". Despite is a word for contrast. We see that in the first part, the senator is not willing to cooperate (hardline), but in the second part, she has a "concessionary tone" which means she actually ends up cooperating.
Hence the sentence is basically saying that the senator said she would not cooperate, but in reality, she cooperated anyway.
I will expand the answer based on your comments