Good answers, but the OP asked for "analogies without mathematics".
First, convolution is a operation, just like addition (you've picked two apples, you pick two more) or multiplication (you fetch a bucket of water five times to fill a larger container). So, it's not a matter of whether it's necessary, the computation is something that lets us explain or predict this property.
Consider yourself in a large empty room, perhaps a gymnasium. Clap your hands together once. If you were in a wide open space outdoors, this would produce a very short sound, but here it might cause a sound that lasts several seconds.
You know this is due to reflections of the clap as the sound bounces off the wall, then those reflections continue to bounce off walls. (At some point the energy falls below a threshold and we consider it stopped.)
But a more precise way to describe it, and one that lets us better predict the resulting sound of a different source (perhaps a violin) in that room would be to describe it mathematically as a convolution.
For instance, what would happen if we clapped twice, a half-second apart, the second clap at half the loudness of the first? The resulting sound would be the same as the first, plus a copy of the first delayed by a half-second.
Going back to the single clap. at one time in their lives, many people have learned the character of an large unfamiliar room by clapping (or maybe shouting a truncated "hey"), and listening to the result. Crudely, they are taking the "impulse response" of the room. A sharp clap is not a bad impulse, really, it forces an impulse of air (but get a good slap of the hands—cupping them muffles it a bit—you need a flat frequency spectrum). Firing a starter pistol or popping a ballon works too.
If you record that impulse response, and convolve it with the sound of the that same (ideal) cap outdoors with no reverberation, the result is the same sound you got in the room. OK, that doesn't buy us anything, it's the same as the impulse response we recorded. But what if our source sound is a sung "laaaaaa!" (recorded dry), and we convolve that with room's impulse? It sounds exactly the same as if that person sang the note in the room.
The why of this is easier to understand if we talk about digital recordings. Each sample is an impulse, at various amplitudes. Somewhat like many claps at a continuous high rate. So, you could go back to the idea of the two claps that cause overlapping and scaled copies of the room's impulse response.
That's what the math of convolution gives us. If we determine the impulse response of a room, we can easily calculate what any stationary sound would sound like in that room (at least at that specific point). A drum-strike, a elephant bellow, a mouse squeak, an acoustic guitar performance.
Some more non-mathematical explanations here: https://www.earlevel.com/main/2012/03/05/convolution—in-words/