Black should exchange rooks after Re1 because you simplify the game while having a winning position. Every piece exchange generally favors the player that's ahead.

After the exchange Black can play h6 to force 2 of white's pawns onto the H file where they are no threat whatsoever because there is no way for white to guard the queening square, so black can simply plant their king there and indefinitely hold against 2 pawns while their rook bullies the bishop and the king.

My assumption is that white blocks E1, then black moves the other rook up to the 2 line, which blocks the white King from moving down to the 2 line. No other piece can do anything about it, and the king is not close enough to attack if white takes the black rook on the 1 line and is blocked from the 2 line. From there it depends on black trapping the King in a mate. Unless there's not something I'm seeing.

What if rather than moving the rook, black queen moves to D6?

If it says "checkmate in 3" its an actual checkmate, idk why but "winning" in puzzles means just "a big advantage"

EDIT: Yeah, I just downloaded and went through all the puzzles. Its the first one that isn't a checkmate, you just win a big trade. Typically the harder puzzles aren't about finding a mate always, sometimes it's winning a piece or just getting a good fork/pin/position.