Practice of saying Sadhu! Sadhu! Sadhu!

Mahasi Sayadaw gives some insight, if not an actual answer in his discourse on the Hemavata Sutta:

The Buddha was constantly into the jhāna, and for that He is adorable. While, after the end of a part of a sermon the audience exclaimed in one voice, "Sādhu! Sādhu! Sādhu! (Well done!) the Buddha went into jhāna even during that brief interval. And then He resumed the sermon. Such constancy is really marvelous."

There are occasions for the audience to say "Sādhu" during my preaching but they are rather few. But in Myanmar it is usual for the audience to say "Sādhu" at the end of a Pāḷi gāthā (verse) of which the preaching monk gives a literal translation. When the monk ends in a long drawn-out voice with the (Myanmar) phrase "phyitkya le dawt tha dee" the audience says without any hesitation, Sādhu. They don't care to notice whether the verse so recited and translated relates to a subject which calls for an exultant hailing or not. They just note ending words "tha dee" and drone out "Sādhu."

For instance, in the Vessantrā jātaka, king Vessantrā gave away his two children, a son and a daughter of tender ages of four or five, to Jujaka Brāhmin. The Pāḷi verse in that connection describes the Brāhmin's cruel treatment of the children who wept desolately; how the Brāhmin beat them cruelly and dragged them away. When the preaching monk recited that verse and translated it into Myanmar and ended his version with the usual "tha dee" the audience droned out the usual "Sādhu". Well, that is the part of the story which calls for sympathy and sadness from the listeners, not exultation, and so the "Sādhu" went awry. But in Myanmar the audience don't care to discriminate.

In Sri Lanka, however, the audience intones "Sādhu" three times only for the part of the sermon which related to attainment of Arahatship or Nibbāna, for that is an occasion of exultation when a congratulatory note of joy, such as "Sādhu", is called for.

During the time of the Buddha the practice of saying "Sādhu" must be of the Sri Lanka pattern. When the audience said "Sādhu" three times, the Buddha paused, and during that brief interval He went into jhāna, and soon after the saying of "Sādhu" by the audience, He resumed His sermon, He never remained idle, How adorable!

The preaching monks of today may not be entering into the jhāna; that brief interval is probably the time of resting his voice or it is the time for him to think of the words he will utter when he resumes his sermon.

It was certainly common in the Buddha's time to respond to an appreciated statement with the word "sādhu"; this occurs throughout the tipitaka and commentaries. E.g.:

“aparopi ce, cunda, sabrahmacārī saṅghe dhammaṃ bhāseyya. tatra ce tumhākaṃ evamassa — ‘ayaṃ kho āyasmā atthañceva sammā gaṇhāti byañjanāni ca sammā ropetī’ti. tassa ‘sādhū’’ti bhāsitaṃ abhinanditabbaṃ anumoditabbaṃ; tassa ‘sādhū’’ti bhāsitaṃ abhinanditvā anumoditvā so evamassa vacanīyo — ‘lābhā no āvuso, suladdhaṃ no āvuso, ye mayaṃ āyasmantaṃ tādisaṃ sabrahmacāriṃ passāma evaṃ atthupetaṃ byañjanupetan’ti.

‘But, Cunda, if you think he has got the right meaning and expressed it correctly,... you should say: ʺGood!ʺ and should applaud and congratulate him, saying: “We are lucky, we are most fortunate to find in you, friend, a companion in the holy life who is so well-versed in both the meaning and the expression!”

-- DN 29 (Walshe, trans)

As for the tradition to repeat it three times, I can't find an example in the tipitaka; twice seems common - dual repetition in pali is certainly a common construction (e.g. punappuna.m - again and again):

“sādhu, sādhu, puṇṇa! sakkhissasi kho tvaṃ, puṇṇa, iminā damūpasamena samannāgato sunāparantasmiṃ janapade viharituṃ. yassadāni tvaṃ, puṇṇa, kālaṃ maññasī”ti.

“Good, good, Puṇṇa! Possessing such self-control and peacefulness, you will be able to dwell in the Sunāparanta country. Now, Puṇṇa, it is time to do as you think fit.”

There is mention in the commentaries (e.g. DN-A) of the phrase "tikkhattuṃ sādhukāramadāsi" ("giving a sādhu-making three times") but I am not clear if this really means saying sādhu three times. In modern times, this is certainly how the Pali phrase is used and understood, but we also see (Jat-A 540):

“sādhu sādhu, bhikkhū”ti tikkhattuṃ sādhukāraṃ datvā...

having given sādhu-making three times, [saying] "sādhu, sādhu, bhikkhu!"...

Anyway, it's clearly a part of the tradition since ancient times in one form or another.

As to why it is considered beneficial, rejoicing in the the goodness of others is one of the ten means of acquiring goodness.