A hand I could have won, but should have lost

Thoughts?

PokerStars Game #4262433054: Hold'em No Limit ($0.50/$1.00) - 2006/03/12 - 12:39:51 (ET)
Table 'Roberta' Seat #1 is the button
Seat 1: XianZe ($113.35 in chips)
Seat 2: fieryhex1 ($105.55 in chips)
Seat 3: Nuke11 ($65.55 in chips)
Seat 4: joshpytt ($99 in chips)
Seat 5: Demiparadigm ($98.35 in chips)
Seat 6: jeffstars ($274.20 in chips)
fieryhex1: posts small blind $0.50
Nuke11: posts big blind $1
*** HOLE CARDS ***
Dealt to Demiparadigm [Ah 9h]
joshpytt: folds
Demiparadigm: raises $2 to $3
jeffstars: calls $3
XianZe: folds
fieryhex1: folds
Nuke11: calls $2
*** FLOP *** [4h 9c 7s]
Nuke11: checks
Demiparadigm: bets $5
jeffstars: raises $10 to $15
Nuke11: folds
Demiparadigm: raises $10 to $25
jeffstars: calls $10
*** TURN *** [4h 9c 7s] [3c]
Demiparadigm: checks
jeffstars: checks
*** RIVER *** [4h 9c 7s 3c] [6d]
Demiparadigm: bets $20
jeffstars: calls $20

i dont understand min reraising the flop and then checking the turn. hell i dont really get min reraising the flop.

Value/Information/cheap showdown

i like how you want to play this oop for a cheapish showdown facing action with a hand