When a rower takes a stroke in the water, it is the momentum that the rower exerts that propels the boat to move. Momentum can be explained by Newton's Third Law, which states that every action has an equal and opposite reaction. This law can be applied to rowing in the following manner - the oar moves the water back and thereby causes the boat to move forward in the opposite direction.

Momentum = mass x velocity

The momentum that the rower exerts in the water will be equal and opposite to that of the momentum that the boat gains in return. This movement is most obvious in the first stroke when the boat is initially at rest with a velocity = 0. However, when the boat is already moving and a normal stroke is made, the transfer of momentum is not as easily discerned because the water has been disturbed. But under closer observation, it can be seen that the water does indeed move backward as the boat moves forward. Let's break down the system before and after the stroke:

V = velocity
M_b = mass of boat
M_w = mass of water
 

The momentum before and after the stroke must equal to 0 since the total momentum cannot change. Newton's Second Law states that the rate of change of momentum is proportional to the Force applied.