Consider a swing, like the kind found at a playground. If you sit on it and shift your weight forward and backward at just the right rhythm, you can get the swing to go very high.
It goes high because the combination of the swing and the mass of your body is resonant at a particular frequency. When you shift your weight to "pump" the swing, you add just a little more energy to the swing. And when you pump at the right time, this extra energy is added to the stored energy from all the previous pumps, so with each swing you go a little higher than the last one. But this only works if you pump at the resonant frequency.
If you pump at some other frequency, you just jiggle around a little bit. You don't go higher and higher, because the actions of each pump don't reinforce each other.
Imagine you are swinging along happily, and simultaneously you receive a phone call, and your phone in your pocket is on vibrate. The vibration from your phone is also a shifting of weight, just as you are doing to pump the swing. But it is at a much higher frequency. Does it alter your motion on the swing? Technically yes, but the effect is very small because the vibration is not at the swing's resonant frequency. Imagine any perturbation you like: perhaps another person on the swing with you, but pumping at some other frequency. These actions might alter the swinging motion a little bit, but the swing responds most significantly to its resonant frequency, even if there are other oscillations going on at the same time.
An LC filter is a resonant system, like a swing. The difference is a swing involves an oscillation between gravitational potential energy (at the top of the swing) and kinetic energy (at the bottom of the swing), whereas an LC filter oscillates between energy stored in the electric and magnetic field of the capacitor and inductor respectively. The LC filter will respond strongly to oscillations at its resonant frequency, while other oscillations at other frequencies have only a negligible effect.
All modulations, not just FM, can be considered a "range" of frequencies. The only signal which is exactly just one frequency is an unmodulated carrier, which contains no information and so isn't used for communication. Some modulations use a wider range of frequencies than others, but no practical modulation uses a range of zero width.
That said, how can a filter work when the signal consists of a range of frequencies?
Real filters, even swings, have a resonant frequency where they are most sensitive. As the frequency deviates above or below that resonant frequency, the filter response diminishes, but it does not immediately drop to zero. The objective in designing a filter for a radio is to design a filter which passes the range of frequencies allocated to the signal, but no more. A very simple filter, like one made of a single inductor and capacitor, is "good enough" for some applications. But a radio designed for performance rather than simplicity will have more complicated filters, with more than one inductor and capacitor, to make the filter perform better. There are often multiple stages of filtering. Filtering, whether analog or digital, is a significant part of designing a radio, and a complex topic in itself.