The actual statistics used to test reliability can be quite complex. However, the ideas are simple and are just forms of correlation and regression.

Say you have a test that measures a personality trait. You would like all the items to give you consistent information about the trait.

How could you do that?
Here's a clever idea - let's take half the items, compute your score and take the other half of the items and compute a separate score for each. If you found a high Pearson's correlation coefficient between these split halves then it would look like the two parts of the test agree with each other. The whole test would seem to have good internal consistency or reliability.