1)Why do excited atoms give off a bright-line spectrum instead of continuous spectrum?

Answer 1

Because the jumps are exact and only these jumps are possible.
In the case of Sodium lamp, for example, the electrons are excited to a higher energy band and emit light when they come back down to the "ground" state. This jump is well quantified and will give the yellow light of the sodium lamp.
Was that your question or were you asking why there are bands and why are they quantified?

Answer 2

Well, here's the long version, anyway... Atoms, or rather, the electrons around atoms, only emit a definite spectrum because these electrons have only certain levels they can "jump to" as they orbit the nucleus. An electron needs a certain amount of energy to get from level one to level two, for example, 13.6 eV (electron volts, a unit of energy) and more energy will cause it to miss the first energy level and "fall back" and not enough energy will cause the electron to only get halfway or a third of the way there and then fall back to where it started. There aren't "in between" levels, at least not as we've observed an atom. Certain amounts of energy produce the quantized spectrum instead of the continuous spectrum because not every energy amount (e.g. 5.39847 eV) will provide an electron with the energy it needs to jump energy levels. And while 13.6 eV may get an electron from level one to level two, it'd be too much to go from level two to level three. (An electron can get energy by either being induced by an outside electric/magnetic field or by gaining energy from a photon which collided with it.)

Answer 3

Quantum mechanics. The energy released by an excited electron returning to a lower energy state is exactly equal to the difference in energy levels. These discrete energy levels (quantum levels) are unique for particular electrons in particular types of atoms (elements). Therefore, we know the chemical composition of distant stars because we can see the characteristic spectra from the atoms that were excited on the surface of the star. You may be interested to know that atoms emit characteristic radiation that is outside the range our eyes can see, wavelengths above and below the colors of our rainbow. I hope this helps.