In English, the word "hero" can mean two slightly different things. As you're using it, it's a person who has performed some great feat in service to another, especially if that feat involves sacrifice. A hero in this sense is performing a selfless act.
But that meaning is a relatively modern extension of the original idea of heroes being "just" people who perform great feats. If you read the great epics, any people the heroes end up saving are often incidental. The heroic acts are for the glory of the heroes, a distinctly selfish act.
It's a bit difficult to explain, if you don't know the feeling. There's a reason I said "been through" rather than "done". Things can happen so quickly that you don't really apply rational thought to what you're doing. And as with all such situations, we tend to post-hoc rationalize what we did. So for a heroic[selfless] act to not be a heroic[selfish] act, it must be done solely for the good of those involved, not for any praise or glory for the "hero". It can make the act feel meaningless to receive praise for it.
You're often the clearest eye-witness to the event, but that doesn't mean you're objectively reliable. At best, you don't know why you did any particular thing. You might say "anyone would have done it." It feels bad to receive praise for a common act.
At worst, you second guess everything. You worry about the mistakes more than anything "right" you did. What if you hadn't hesitated for what felt like an eternity (when outside observers would probably say you sprung into action immediately)? What if you hadn't forgotten to carry your pocket knife today, so you could have cut the seat belt into a tourniquet (despite seatbelts being extremely difficult to actually cut). What if you hadn't pressed so hard during CPR that you broke their ribs (except CPR done correctly is very likely to break ribs)[0]? In emergency situations, the odds are against you, so even if you do absolutely everything right, you're probably not going to "succeed". It definitely feels bad to receive praise when you think you've screwed up.
That's why you'll see so many people say they "just want to move on." It's easier to bury the issue than deal with any one of those things, and it's highly likely the person is going through all of them. The most thanks someone will want is to see the person they tried to help walk away happy and safe; a thanks from that person will be sincere and will feel real. From anyone else who wasn't involved, it will only serve as a reminder of something the person would rather forget.
People don't want to receive praise when they don't feel like they deserve it. People feel like they deserve praise when they've done hard work and they've done it well. You want to thank a cop or EMT or doctor or search-and-rescue volunteer? Don't thank them for that one event over which they probably feel conflicted. Thank them for their choice to join their profession and their choice to continue to wake up every day and go back to it.
Incidentally, that's also why you probably shouldn't thank a soldier "for their service." You don't know why they joined. Some people have joined not out of a sense of service to country, but a sense of desperation to get out of a bad situation at home. And once they have joined, they can't easily un-join. Continuing to show up to be a soldier is not a daily choice. Or maybe they've joined for all the "right" reasons and they believe every day that their choice was the right one, the vast majority of military personnel are in support--not combat--positions, and that person might feel they are not deserving of praise because they are not in danger. Thanking a soldier for their service--especially if you're a random person on the street and only know they are a soldier because they are wearing a uniform--could be reminding them of lots of things outside of their control.