Acid / base indicators need to be weak acids or bases because they need to undergo a reaction just after the equivalence point. For an indicator to be appropriate, you need the pH at the equivalence point (which is determined by the nature of the salt formed) to be near the pKa of the indicator.
For example, with acetic acid / sodium hydroxide, the salt (sodium acetate) is a weak base. The pH of the equivalence point is around 9, making phenolphthalein (pKa = 9.4) to be an appropriate indicator.
If I represent phenolphthalein as HInd (in its acidic form) and am adding NaOH from the burette, then the colour change is due to
HInd + OH- <---> Ind- + H2O
where HInd = colourless but Ind- = pink
You will get some pink colour before the end point as added hydroxide can react with the indicator or with acetic acid, depending on which is encountered first. However, any phenolphthalein converted to its conjugate base form will be rapidly converted back to its acidic form by acetic acid:
CH3COOH + Ind- <-----> CH3CO2- + HInd
This equilibrium lies far to the right as the acetic acid (pKa = 4.76) is a much stronger acid than is phenolphthalein (pKa = 9.4) even though both are weak.
So, it is only once all the acetic acid is consumed (at the equivalence point) for the addition of a tiny amount more hydroxide to permanently shift the equilibrium between HInd and Ind- to favour the coloured conjugate base form and thus for the end point to be found.