A watt (or kilowatt or megawatt) is a measure of power. Power is the rate of energy transfer. For example, a watt is defined as 1 joule per second, where you can think of a joule as one nicely measured packet of energy.
Watt-hours (or kilowatt-hours or megawatt-hours) is just another way of measuring energy. A kWh is one kW of power flowing for one hour, which is 1000 joules going by every second for one hour. Since there are 3600 seconds in a hour, 1 kWh is therefore exactly the same as 3.6 Mj. The electric grid deals with large power levels and large energy transfers, so that discipline has evolved to express energy in MWh and kWh because that is more directly relevant to how they transfer and store energy.
I didn't look at your link, but what I remember from the Beacon Power flywheels is that each one is rated for 100 kWh. The 100 kWh is stored by spinning large and heavy flywheels in vacuum on magnetic bearings. During normal operation, the flywheel speed goes from its maximum to half that, which means from full to 1/4 of the flywheel's total energy. If I remember right, the 100 kWh figure is the difference between the two normal operating extremes.
I think the discrepancy between 20 MW and the 100 kWh of individual flywheels is that they are referring to a large plant with many flywheels.
25 kWh transferred over 5 minutes is 25 kWh transferred over 1/12 hour, which is 300 kW continuously for 1/12 hour. So yes, your last sentence is correct.