Why care about truth?

Because it allows us to make sense and is preserved by logical implication.

$$(\mathrm{False} \implies \mathrm{False}) = \mathrm{True}$$

$$(\mathrm{False} \implies \mathrm{True}) = \mathrm{True}$$

$$(\mathrm{True} \implies \mathrm{False}) = \mathrm{False}$$

$$(\mathrm{True} \implies \mathrm{True}) = \mathrm{True}$$

Note that False implies anything (hence if someone is lying to you, you stop listening) and that True cannot imply False as long as valid inference rules are used. So truth is important because of modus ponens, the ability to chain statements.