Why do indicators change colour at convenient points during a titration?
Because indicators are carefully chosen to behave in this manner.
An indicator molecule is a large organic dye whose acidic and basic forms have distinctive colours. Usually you can discriminate between the colours when there is only a 10-fold difference in the concentration of the acid versus the base. In a titration this usually corresponds to less than
Under the basic regime, the indicator has a specific colour, and under the acidic regime the indicator has another. The colour change, which signifies equivalence, is VERY rapid, and easily discriminated.
When you do titrations in the lab, it is always a good idea to titrate to the POINT, i.e. the onset, of colour change, rather than to the colour change itself. Anyway, listen to your demonstrator in the lab, and he/she will have something to teach you.