Hi,
As you're using the 4.2.0, you have the first version of HikaShop we released with the new AJAX system for filters.
So there are two things first:
1. I don't see the AJAX system extra HTML attributes on your website when I check the HTML. So it's probable that you have some view overrides in your template which could lead to this. Without these HTML attributes the AJAX refresh of the listing won't work properly.
2. We've fixed a lot of bugs on the filter system since the 4.2.0 so updating might be a good idea.
3. However, from what I can see, the main issue here is that you have turned off the "auto submit on click" setting of your filter and you don't display the "filter" button either, so once the dropdown is changed, there is no way for the user to act upon the change to see the new listing. So the first thing I would recommend is to check that setting of your filter.