What is significant about the relative concentrations in solution when half the base to the equivalence point has been added?

At the equivalence point, the reaction of the acid and alkali is complete as an amount of alkali equivalent to the acid has been added. Half-way to the equivalence point, half of the acid has reacted with NaOH.

Therefore, [HA] = [A^-].

How does this half-way point help to determine the Ka?

We know that [HA] = [A-] at half-way to the equivalence point and we can read off the pH.

Ka = [H+][A] / [HA]  where [H+], [A] and [HA] are all equilibrium concentrations.

Ka = [H+] [A] / [HA] and since [HA] = [A-], Ka = [H+]  

Taking logs of both sides pH = pKa

The pH, and hence the pKa, can be read from the pH curve plotted.

How might this experiment be caarried out without a pH probe?

An indicator solution such as phenolphthalien could be used.