The grammar does the cutting
Ya is a cutting particle with no English equivalent. It does not mean anything; it stops the sentence in place, so that furuike sits alone for a beat before the frog arrives. Every translation that renders ya as a comma loses the pause; the dash is a better approximation because it keeps the silence visible on the page.
The second line has no subject marker and no tense marker that English would require. Kawazu tobikomu is roughly frog / jump-in, and whether it is present or past, one frog or several, is not stated. The translation has to decide, and every decision adds information the original withheld.
Mizu no oto is not a metaphor
The poem ends on a sound, not on a meaning. Water's sound. There is no word for quiet, stillness, solitude, or enlightenment anywhere in the seventeen sounds. The stillness is produced by the reader, who notices that after the splash the poem stops without describing anything else.
This is why the poem resists paraphrase. Say what it means — a sudden sound deepens the silence — and you have added the very statement the poems refuses to make. The refusal is the technique.
The two-part structure, and what the reader supplies
The poem is built from two elements: the old pond, which is a state, and the frog's entry, which is an event. They are separated by the cut and not joined by any connective. The relation between them — that a small sound proves the pond was silent, that age and suddenness are being set against each other — is entirely the reader's contribution.
Bashō wrote a number of versions of this poem over several years, and one earlier draft had the frog singing rather than jumping. Changing the action from singing to entering removed the last trace of a human frame: a singing frog is charming, a frog going in is just an event. The final version is colder and much better for it.
A test you can run on your own drafts
Take a short poem of yours and delete every word that tells the reader what to feel. If nothing readable is left, the poem was made of commentary. If a scene is left and it still moves, you had a poem underneath the commentary, and you have just found it.