As a moderator on parenting.se, I thought it might be helpful for me to share our experiences with this rule.
That clause was part of our original faq from day one. From our perspective, adding it was intended to reduce answers that are essentially "I don't have any experience with this, but this is how I think it would work". Those types of answers are frequently of very low quality, and they can be a fairly persistent problem.
Ideally, we prefer answers that cite actual research and references. That isn't always possible, depending upon the question. In the cases where research isn't available, direct experience is usually a welcome substitute.
After all, the stackexchange sites are intended to be a repository for expert answers.
If an expert says "this is how I handled it", the answer is generally useful, whereas if the answer is pure speculation, even if the speculation comes from an expert, it tends to be much less useful.
There seems to be a perception here that the wording used somehow encourages low-quality answers. This has absolutely not been my experience. People don't look at the faq, read that, and start posting "me too!" answers that don't provide useful information. The users who do that generally haven't read the faq in the first place.
Rather, what we've found is that including that clause in the faq gives our community a point of reference to help explain to posters why certain answers are considered low quality.
In my nearly 10 months of moderating at parenting.se, I've never seen that section of the faq used to justify a low-quality answer. Instead, what typically happens is it is brought up by way of explanation to a new user, to explain why their answer is being down-voted, or to help show them what the answer needs to be improved. I've seen both moderators and active members of the community cite the clause from the faq, and it has always (to my recollection) resulted in a positive outcome (future answers from that user tend to improve, and sometimes they will go back, edit their answer, and provide useful sources and references).
It is worth mentioning that we do not enforce this rigidly, nor have we ever had any reason to. We do have plenty of high quality answers that do not reference direct experience or external research and sources. Those answers continue to get upvoted/accepted, and no one has ever said "this is a good answer, but I see no evidence that it meets our criteria for the 'back-it-up' rule."
I believe strongly that it will benefit you as a community to have the clause included in your faq from the beginning.