I was very lucky because as a child I sang a lot of gregorian chant in the different modes, and it's definitely helped to understand the system, because I had a hands on experience from a certain age.

I'll have a bash at explaining it as I would do if I was sat down at a piano, which isn't necessarily the best way to explain in text, but it's the easiest way I know.

A major scale (and major is effectively Mode I, the Ionian mode) is a certain progression of notes - you start on a given note (say C, or D, or whatever) and you go up in a set order: up a Tone, up a tone, up a semitone, tone, tone, tone, semitone. All major scales are that same progression. So if you're playing within a major key, that basically means that there's a pool of 7 notes that you're drawing from, and all of those notes have a certain relationship with one another. So if I write a tune in a major key it has a particular feel.

Ok, now let's imagine that the notes I'm using AREN'T within a major scale, so don't come out of that set series of starting note, up a tone, up a tone, semitone, tone, tone, tone, semitone.

Let's say that the scale starts on a given note and then goes up tone, semitone, tone, tone, tone, semitone, tone.
If you listen to that, you can hear that you've got a run of notes with a slightly different feel to the run of notes you'd expect from a major scale. If you're picking notes only from within this series of notes, then you're working in Mode II, the Dorian mode.

Or you start on a given note and then go on a run of tone, tone, tone, semitone, tone, tone, semitone. If you work only with the notes in that series, you're in the Mode IV, the Lydian mode.

It's all about the different permuations of sound you can create between different notes. If you're only working in major or minor, you're missing out on whole different kinds of sound.