Eco-roof flow meter with "counter channel" output - what is that?
Your flowmeter is a tipping bucket - it dumps when a fixed volume, N, has accumulated.  When it dumps, it momentarily closes a reed switch.  Thus every time the switch closes, you've amassed an additional N of total flow.  By taking the first derivative of the output - dividing by time - you can compute the flow rate.

These reed switch closures with each dump can be made into electrical pulses if the switch is made to complete and break a circuit.  If you count the pulses over eternity, you can compute the flow over eternity.  If you count the pulses per unit time, you can compute the flow per unit time.   

To interface with your datalogger, you'll want to use the "pulse" input.   A pulse input is basically an input to a digital counter inside the datalogger.  Every time your bucket dumps, the switch closes and the datalogger will register a count event, incrementing the internal counter by one.  The datalogger hardware and software will determine exactly how counts are managed, whether they are simply counted or differentiated to give a rate.  You'll need to look into the configuration docs for your specific model to adapt it to your specific needs.

What you have is similar to a digital output - the switch is either open or closed.  The total flow is a function of events over time.  An analog output would supply a continuously variable signal (voltage, current, or frequency, typically) that would resolve to engineering units in your datalogger.  For example, a pressure transducer might provide an analog voltage output ranging from 0-5V as the pressure ranged from 0-500 PSI.  Your datalogger would read the voltage directly and convert it into a digital number, which would then be scaled back in to the 0-500 PSI engineering units by your datalogger software.

The good news is that this is all pretty safe stuff (i.e. hard to damage either the equipment or yourself).  Try connecting the flowmeter to your datalogger on your desk.  Simulate water flow by tipping the bucket and verify that it shows up the way you expect in the datalogger software.