When you tell someone to tread lightly, you're saying that they should be careful about how they phrase what they're about to say.
Tread is a word that comes from Old English, meaning "walk or step." If you were walking over a rotted wooden bridge, you would certainly look very carefully before you stepped, and try to make your footsteps as soft as possible — you would tread lightly. Similarly, being told to tread lightly in a conversation is way of expressing that the topic you're talking about is sensitive and requires tact.