It might be easier to imagine them as phasors, little stopwatches where the second hand rotates around some integral number of revolutions -- the value of k dictates how quickly the phasor hand rotates. Essentially, what it's doing is correlating your signal with a harmonic basis vector, and the number of revolutions of the phasor goes up by integral amounts for each basis -- the first one makes one full revolution over the transform period, the second makes two revolutions, etc.