What does \$x^*(t)=x(t)\$ mean in signals and systems?
\$x^*(t)\$ is the complex conjugate of \$x(t)\$

So, when \$x(t)\$ is defined as $$x(t)=a+bj$$ then $$x^*(t)=a-bj$$.

When $$x^*(t)=x(t)$$ it means

$$a+bj=a-bj$$

This is only true when \$b=0\$, so \$x(t)\$ has to be a real number.