Degrees of freedom "The number of degrees of freedom is a measure of how certain we are that our sample population is representative of the entire population - the more degrees of freedom, usually the more certain we can be that we have accurately sampled the entire population."
Can someone please explain this statement?
 A: There is a sentence prior to the passage quoted by the OP that I believe helps to interpret this:

In statistics, the number of degrees of freedom (d.o.f.) is the number
  of independent pieces of data being used to make a calculation. (...).
  The number of degrees of freedom is a measure of how certain we are
  that our sample population is representative of the entire population
  - the more degrees of freedom, usually the more certain we can be that we have accurately sampled the entire population.  

So here 
"more degrees of freedom"$\equiv$ "greater number of independent pieces of data"  
This starts to sound familiar, since it points to the size of a sample of independent draws from the population. Moreover, on focus here are experimental data, so all nice properties I guess are assumed to be guaranteed, and therefore the larger the sample size of independent pieces of data, the more strongly the consistency property of estimator will actually emerge and reflect upon the estimates obtained.  
So it appears that, for the author of the passage, the logical chain here goes as follows:

"more degrees of freedom"$\equiv$ "greater number of independent
  pieces of data"

and 

"greater number of independent pieces of data" $\Rightarrow$"greater
  accuracy in recovering the population moments from the data"

and

"greater accuracy in recovering the populations moments from the data"
  $\Rightarrow$ "the more representative of the population is the sample"

So 

"more degrees of freedom"$\Rightarrow$ "the more representative of the population is the sample"

It appears therefore that the author of the passage employs the term "representative sample" with the meaning of "miniature population" or of  "typical or ideal case", to follow the typology of Kruskal and Mosteller as re-relayed here.
