A "trophy" is the shiny little statue or cup award to an athlete who's won a competition. A trophy is a symbol of winning. A "trophy house" is a house built, not for comfort, but to show people that you've "won." It is important that it look big, expensive, and impressive.

"What is a trophy house?" Just a tip: "met" is not the right word here. We normally use "encountered", "discovered", "found" or "came across". Using "met" sounds as if you said hello.

In this case, a "trophy house" is a big, fancy house that you have, mostly to impress other people. To show you are rich or an important person. You may not need all the space or the facilities, but you live in this house anyway to display wealth/social importance to other people. It's a "trophy" that you've won with your money or hard work etc.