The term "qualify" can be used in different ways, so a reference to a specific question number would be very helpful here, otherwise we're just guessing at the particular usage you mean
That said, "qualify" as often used on the LSAT means to modify, limit, or alter. So, a "qualified commitment" to something would be a less-than-complete commitment, or a commitment that has some stipulation or condition attached to it. "Qualify an assertion" would probably mean to change it or explain some part of it in a way that limits it. Again, depends on the exact context around it.
By the way, the other LSAT use of "qualify" is the standard "approval" or "certified" sense, as in "she is qualified for this job." That's not the use in your question stem, I'd wager, however.