There are three ways to get out of check (including checkmate). You can...
- Run away
- Block the check
- Capture the checking piece
There are two checks here (pawn and rook). If the king runs to either empty square, the bishop (and in one case, the rook) can capture. If the king captures the pawn, the knight can capture. That takes away #1.
You can block the rook check (with the queen or bishop) but not the pawn check. That takes care of #2.
You can capture the pawn (with the rook or queen or king), but you can't capture the rook, so that takes care of #3.
So, it is checkmate.
If the white pawn wasn't checking, or if the white knight couldn't recapture, it wouldn't be checkmate (you could block the check or take the pawn with the king). If the white rook wasn't checking, it wouldn't be checkmate (take the pawn with the rook or queen). It takes both checks in this case to produce checkmate.
This is a rather complicated checkmate. In my experience, most beginners would not be able to understand it, nor would they be able to find the move that produced it (the pawn capturing something to give check, which also produces a discovered check with the rook). Don't give up on the game because the app gave you a rather nasty position. Most real life chess is simpler than that.