This is a pretty clear indication that your filter is clogged and needs to be replaced. HEPA filters are different then say pre-motor filters, or sponge filters. Because they are usually the last filter before the air is exhausted, it's usually catching very tiny microscopic particles. Because of this, trying to determine if the filter is dirty or clogged can be difficult, if not impossible on HEPA pleated exhaust filters.
The fact that your filters continues popping out, indicates that the air is not being able to pass through the filter as it's supposed to, and is instead the air is having to try and find a different way to escape, causing the force of the back up of air to blow the filter out.
I think if you replace this filter, it should fix the problem completely. Because this filter is designed to capture microscopic particles rather then large particles, washing this filter, or getting it wet is not advised, and wouldn't be an effective way to clean it. You really will need to replace the filter soon, but if you'd like to try and prolong the life of your filter, you may use an air compressor, and blow from the top of the filter (the area that looks clean), and clear the microscopic particles, and dust out of the filter this way. If you suffer from breathing issues, asthma, or allergies, I would highly advise against doing this without a mask on. The dust that will come out is extremely fine dust that can cause people with the above conditions to suffer greatly. Doing this outdoors is highly recommended.
An air compressor can certainly prolong the life of HEPA filters, but again, sooner or later you'll need to replace the filter.
If you're in need of buying one of these filters, you're in luck! We carry these filtes on our site, and they are in stock. Here's a link back to the filter on our site if you need to order one. We offer free shipping, 60 day returns, free return shipping, and no restocking fees. Please feel free to reply to this solution by using the comment box at the end of this solution. If you found my solution to be helpful, please be so kind and rate my repair, or leave feedback for me. This is greatly appreciated by the FixYa community, and is really what keeps repair professionals like myself to keep answering free questions, and not just the premium questions which we get paid to answer.
Thanks so much,