Castling developed in incremental steps with significant regional variation. Cessole, a Dominican Monk in the 13th-14th centuries, wrote in his book Liber de moribus Hominum et officiis Nobilum ac Popularium super ludo scacchorum that once per game, either player could move their previously unmoved king (say on e1 for white) in one move to b1, b2, c1, c2, c3, d3, e3, f3, g3, g2, g1. If you're having trouble picturing that, here's a diagram. Cessole, an Italian (who once headed the Inquisition in Genoa), notes there are many restrictions and regional variations.

Our next evolution in castling is noted in Luis Ramírez de Lucena's famous book published in 1497: Repetición de Amores y Arte de Ajedrez con 101 Juegos de Partido. Lucena describes castling occurring as two separate moves. First the rook moves (like from h1 to f1). Then on a subsequent move, the King can leap to the other side (e1 to g1). Here's the algebraic notation showing what this might look like: 1 e4 e5 2 Nf3 Nc6 3 Bc4 Bc5 4 d3 Nf6 5 h3 d6 6 Bb5 a6 7 Ba4 Rf8 8 Nc3 Kg8 (the leap) 9 Be3 Bxe3 10 fxe3 h6 11 Qd2 Qe7 12 Rd1 Be6 13 Kc1 (the leap).

Our final evolution is noted in Rodrigo (Ruy) López de Segura's book published in 1561: Libro de la invención liberal y arte del juego del Axedrez, where this maneuver is consolidated into one move. That said, widespread adoption of this rule took hundreds of years, notably in Italy. Generally, castling is seen as a defensive move, as the king is typically exposed in the center of the board. Players are taught to do the ABCs in the opening: Activate pieces, Build a center, Castle. Here's a fun game showing how to attack an exposed king.

Regarding en passant, we're not quite as sure how this developed. We believe that in the 13th century, rules were changed to allow pawns to move forward two squares instead of one on their first move, as Cessole writes about this. We also know that in Italy they coined the term passar battaglia, or "dodging the fight", to denote moving a pawn forward two squares to make it adjacent to an enemy pawn, thus preventing capture. From modern reconstructions (try playing without the rule!), we find that this technique can lead to closed positions which severely limit dynamic play, which runs somewhat counter to the idea of speeding games up by allowing pawns to move forward two squares instead of one. We also know that by the 15th century, Lucena has written about the en passant rule. Again, other countries were slow to adopt this (looking at you again, Italy), but by 1880 or so this rule became standard.

For this answer, I've relied heavily on the following excellent resource:

Hooper, David, and Ken Whyld. The Oxford Companion to Chess. 2nd ed. Oxford [England] ; New York: Oxford University Press, 1992.