My question is what does orbits with definite energy mean?
It means that electron circumscribes an ellipse and the total energy of the system nucleus + electron is constant; when the electron is closer to the nucleus, its potential energy is lower and the kinetic energy is higher, but their sum remains the same.
why do the electron in their ground state not emit radiation?
Bohr thought that atom radiating implies that the atom loses total energy and will quickly collapse (nanoseconds). Since by experience he knew it does not usually collapse in that way, he wanted a model where the atom would not radiate - then, in his line of thinking, the atom would not lose energy and the motion of the electron would be stable in longterm.
He probably did not realize that atom can radiate without losing energy on average. Antennae do this all the time.