The Cicada
A careful study of the noise-making apparatus of the cicada can be found in a 1994 paper by Young and Bennet-Clark.$^1$ The authors generated sounds at about 0-16 kHz at peaks on the order of 100 dB using cicadas in various stages of deconstruction. The cicada uses a resonant organ-system called the tymbal which buckles and unbuckles rapidly to produce sound. The buckling-in is caused by muscle contraction and is louder than the buckling-out (relaxation) phase. Air sacs (a feature of many other small noisemakers) serve to amplify the sound. The tymbal itself, for the species in this paper, has a resonant frequency of about 4kHz (Young, page 1017). The song of the cicada as modified/amplified by air sacs and other structures, is often around 10kHz.$^{2}$
Pure Tone vs. Diffuse Tone?
The vocalizations of large vertebrates are a complex superposition of waves that in the frequency spectrum are somewhat spread out. To the extent that a frog or a bird emits a pure tone, the energy will be confined to a narrow frequency range and this may be a strategy for achieving greater amplitude. Given comparable audio sensitivity, however, the intensity of pure tones will depend on amplitude (intensity), regardless of frequency.$^{3,4}$
Other Small Loud Animals
While the songs of cicadas are intense, especially in concert, on an individual level there is competition from other species. According to a Gizmodo article quoting assorted scientists, the snapping shrimp produces a transient snap that is around 200 decibels, a level that one site describes as "deafening." For perspective, dolphins can emit short chirps of 220 dB but these are outside the range of human hearing. The lion emits a roar of 115 dB which is sustained and audible 5 miles away, according to the article. Elephants also are capable of 117 dB cries, as are howler monkeys.
Both the shrimp and the cicada use non-vocal vibration to create their sounds. The shrimp uses a "spring-loaded claw" (the spring is muscle). The localized force of one part of the claw hitting the other generates a bubble (this is known as inertial cavitation). When the bubble collapses it generates a shock wave (noise) that stuns fish (prey).$^5$ The noise of frogs is produced as air passes from the lungs through the larynx, amplified by distended air sacs which resonate. Birds can produce up to 135dB (the Mollucan Cockatoo). They generally force air past (membranes and) a specialized organ called the syrinx located at the bottom of the trachea (see the Wikipedia note on bird vocalization).
Micronecta scholtzi, a 2mm-long aquatic insect, is for its size the loudest known animal. It creates a sound of 99.2 dB intensity which (despite being largely lost in transition from water to air) is audible to humans ashore. According to the Wikipedia note it creates this sound by "stridulating a ridge on its penis across corrugations on its abdomen." The area involved is about 50$\mu m$ across. Details of the mechanism are poorly understood. The article's comparison of the sperm whale's 236 (underwater) dB song gives perspective, as a sperm whale can weigh 14 metric tons.
A Common Aspect of Sound Intensity: Cavity Resonance
Descriptive studies of sound-creation by small animals (with the possible exception of the snapping shrimp) do not fully explain why a one-gram bug can make a bigger noise than a lion. Purely vocal methods of larger vertebrates produce sustained noise on the order of 100 dB but the non-vocal instruments of smaller creatures are capable of short bursts of amazing intensity.
It is difficult to generalize but because air sacs are part of many sound-making schemes, cavity resonance probably plays an important role. Like spring-mass systems or RLC circuits, cavities have resonant frequencies at which amplitude of a signal may be increased (the policeman's whistle is a familiar example). Another paper by Bennet-Clark and Young gives a sketch of a theory along these lines. At resonant frequencies the impedance of the instrument falls sharply and instead of being dissipated (generally as heat) the energy emerges as sound.$^6$
$^1$ Bennet-Clark and Young, The Role of the Tymbal in Cicada Sound Production, J. of Experimental Biol. (1995) 198, 1001-1019.
$^{2}$ The frequency of cicadas is variable, mostly on the order of 10kHz but occasionally very low (< 1kHz), mostly due to body size. See this article.
$^3$ Sound--essentially a compression wave--diminishes with distance. So when the Wiki article on noise levels compares noise levels it includes the distance from the object. For example, 100 db (comparable to the cicada) is the level of noise associated with a jack-hammer at 1 meter away.
$^4$ Audible range for humans is roughly 15 Hz-16000 Hz. As mentioned below, the dolphin can emit very intense high-pitched sounds that we don't hear at all, so the analogy to EM waves (higher frequency = higher energy) doesn't help predict perception. See Pfaff and Stecker, Loudness and Frequency Content of Noise in the Animal House, Lab. Animals (1976) 10, 111-117.
$^5$ M. Versluis, B. Schmitz, A von der Heydt & D. Lohse (2000). "How snapping shrimp snap: through cavitating bubbles". Science 289 (5487): 2114–2117.
$^6$ Bennet-Clark and Young, Short Communication, The Scaling of Song Frequency in Cicadas, J. Exp. Biol. 191, 291-294 (1994).
