Reaction score is just a built-in thing for the community software. It increases as others "like" your posts or comments. I gave you a "like" above.
The trophies are just awards that boost reaction score for certain accomplishments; there's very few of them and mostly occur for new members because they're "posting for the first time" or "receiving their first like" et cetera. Trophies are automatically system given and can't be given by members, or to members.