We used it not for a Disney trip but for a different trip. The day we were supposed to leave, wife got violently ill. Ill spare you the details but if you use your imagination you probably have a 50/50 chance of guessing.
I have trip cancellation on my credit card which is how I booked and paid for everything in advance. They asked for a doctor's note, so when she was feeling better a few days later, she took a quick trip to the doctor, we sent off the note and shortly after that, I was refunded. Its the one and only time in my life I've ever needed to use trip insurance but I'd rather pay for it, have it and not use it, than not have it and wish I did.
Each credit card and insurance company seems to have different ways of making a claim. Like deedubb said though, when you get it, read through and it should tell you what you need to do to make the claim.
In the past I've bought trip cancellation from CAA and Johnson MEDOC as well. Travel agents will also try to sell you health/medical / trip cancellation insurance, but its been my experience what they offer is way more expensive than getting it yourself from somewhere else.
I've said it before in a few posts, but you should check with your / your significant other's workplace. A lot of times employers may have pre-existing deals with insurance companies for this kind of stuff.
I hope some of that helps!