I want to help out my Professor, how do I ask? (Humanities)
Your thesis will be your research experience in the humanities, absolutely.

Speaking quite broadly, in comparison with the sciences, most research is done individually and published individually.  Thus, as your professor has said, focusing on your thesis —which you're doing individually, though with some general oversight from your advisor— is getting you that experience.

Again speaking broadly, consider the process that you've gone through / are going through / will go through for your thesis.


Read a lot of books and articles.
Come up with a really cool idea about something in one or more of them.
Talk to your advisor to decide how you want to approach the topic, maybe getting a handful of ideas of important articles or viewpoints, etc.
Find articles/books/chapters that contribute to your arguments.
Write your thesis.
…
PROFIT! (except for that student debt...)


Now, consider the process for most of us in the field when we write articles:


Read a lot of books and articles.
Come up with a really cool idea about something in one or more of them.
(Optional) Talk to colleagues (either friends, or via conferences) about how you're approaching the topic, maybe getting a handful of ideas of important articles or viewpoints, etc.
Find articles/books/chapters that contribute to your arguments.
Write your article/book/chapter.
…
PROFIT! (except this is academia, so not)


The only real difference is that you're working closely with your advisor, and later in the "real world" (academia), you're going to mainly work solo, except for occasionally consulting with others when you present preliminary work at a conference or when you bat ideas off of friend or colleague's head.  In both cases, though, the actual research work is done by you, the actual planning and development of structure/ideas is done by you, and the actual writing is done by you.

Or consider it another way, what kind of research help could someone give you that would actually be of help to you?  They can't come up with the ideas, because, well, then they get credit.  You don't want them writing it, because then it's not your thoughts and anyways you'd probably end up wanting to reformulate 90% of their arguments/word choice/etc to fit with your exact idea.  At best maybe helping you track down some obscure article that mentions some obscure factoid, but that's almost always going to be inconsequential to your overall development of the topic and hell, half the time when doing those searches is when you find the really good stuff for later articles.

There are some projects that are useful to have multiple people working simultaneously (especially nowadays under the banner of digital humanities), but if your professor needs people, they'll let you know, and as their advisee, you likely be the first student they ask.

tl;dr: Don't feel the need to be useful, just do a good thesis.  That will get you the best experience (and the best letter of recommendation).  If they need help, they'll let you know.