Because they are not three groups of consonants. They are three groups of sounds that no longer exist, and the classes are the fossil they left behind.
Six or seven hundred years ago, Thai had roughly three ways of starting a syllable: voiced (like English b, d, g), plain voiceless (like French p, t, k: no puff of air), and aspirated voiceless (like English p, t, k: with a puff). At that stage Thai had far fewer tones than it does now, and possibly only three.
The sound change
Then the voiced series hardened. Old voiced b became p, d became t, *g became k. That should have been a disaster: hundreds of word pairs that had been distinguished by voicing were suddenly identical.
They were not identical, though. Voiced consonants had always dragged the pitch of the following vowel down a little, and voiceless ones let it sit higher. That pitch difference had been an automatic side-effect. Nobody heard it as meaningful. When the consonants merged, the pitch difference was the only thing left holding the words apart, and speakers started hearing it as the difference instead.
An accident of the throat was promoted to a feature of the language. That is the tone split, and it is where most of Thai's tones come from.
What that means for you
The class of a letter tells you which of the three old categories it belonged to:
Mid class is the old plain voiceless series. High class is the old aspirated voiceless series. Low class is the old voiced series, which is why it is by far the largest and why it behaves differently from the other two.
So when you learn that ก is mid and ค is low, you are not memorising a label. You are learning that ก was always a plain k, and that ค used to be a g. The tone rules are simply the record of what happened to each group.
- ปาpaː
to throw
mid class: was always a plain p, the French one
- พาpʰaː
to lead someone
low class: this was once a b
- ผาpʰǎː
cliff
high class: was always an aspirated p, the English one
The payoff
This also explains something that otherwise looks like a design flaw: why the tone marks do not name the tones. ไม้เอก means 'mark one' and ไม้โท means 'mark two'. They were numbered before the split, when a mark meant one thing regardless of the letter it sat on. Afterwards, the same mark produced different tones depending on which old category the consonant came from.
The mark did not change. The consonants did. That is why you cannot read a tone off the mark alone, and why the class always comes first.