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How does spectroscopy reveal the wavelengths of radiation absorbed or emitted by matter?

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Spectroscopy reveals the wavelengths of radiation absorbed or emitted by matter by measuring the interaction between electromagnetic radiation and the quantized energy states of atoms and molecules.123

The Mechanism of Interaction

Matter consists of atoms and molecules that possess discrete, quantized energy levels. When these particles transition between energy states, they either absorb or emit a [photon]. The energy of this photon ( $E$ ) is directly related to the radiation's frequency ( $ν$ ) by the equation:425
$E = h ν$
where $h$ is [Planck’s constant]. Because the frequency is linked to the wavelength ( $λ$ ) by the relationship $λ = c / ν$ (where $c$ is the speed of light), measuring the energy of the radiation allows scientists to determine the specific wavelengths involved.423

Detection and Analysis

To observe these wavelengths, a [spectrometer] is used to disperse the radiation into an ordered sequence called a [spectrum].61
* Emission Spectroscopy: Atoms or molecules are excited to higher energy states (e.g., via heat or electrical discharge) and then decay to lower states, emitting light at characteristic frequencies.164
* Absorption Spectroscopy: Radiation of known wavelengths is passed through a sample; the detector reveals which wavelengths are absorbed as particles move from lower to higher energy states.164