It’s quite likely that they could predict eclipses, but we don’t have a lot of textual evidence.

The oldest surviving work of Indian astronomy is the Jyotishi Vedanta written by a man named Lagadha around 1,400 - 1,200 BC. The Vedanta were commentaries on the Vedas, explaining some aspect of Vedic knowledge in more detail. Astronomy is an integral part of the Vedas, because there were strict prescriptions on the timing of religious rituals. The Atharva Veda does suggest that astronomers could predict eclipses because it advises that preparations be made to perform certain rituals during eclipses, when they are especially propitious and powerful. The Jyotishi Vedanta also mentions astrologers calculating the timing of both solar and lunar eclipses, but unfortunately it offers no details about the method used.

Before we go ahead, there are a couple of things we need to keep in mind. First, there is no evidence from any of the ancient cultures - Mesopotamian, Egyptian, Indian or Chinese - that anyone understood the exact motion of the earth and moon as early as the second millennium BC. Further, there were no instruments to make precise observations and measure small angles, so any predictions couldn’t have been very accurate. You might have been able to predict the year or month of a solar eclipse, but not the exact date and time. Lunar eclipses could be predicted much more precisely. This difference is due to simple geometry: the moon’s shadow on earth during a solar eclipse is only about 300 km across, while the earth’s shadow on the moon is 12,000 km wide. There’s a lot more wiggle room in predicting a lunar eclipse.

To understand how they did it, we need to understand their calendar. The Hindu lunisolar calendar is mentioned in the Rig Veda, and had been in use in Lagadha’s time (~1,400 BC) for hundreds, or possibly thousands, of years. This calendar is based on lunar months and days, but synchs to the sun periodically by inserting a leap month every 60 lunar months. Like the Greeks, Indian astronomers used a solar zodiac, but they also used an additional lunar zodiac.

The zodiac is the belt of stars and constellations encircling the solar ecliptic, meaning the plane of the earth’s rotation around the sun. During one complete circuit around the sun, a line drawn from the sun to the earth and extending beyond passes through 12 recognizable clusters of stars or constellations, which were the 12 signs of the zodiac: Leo, Aries, Gemini, Virgo, Scorpio, Cancer, etc. In the same way, Indian astronomers had an additional lunar ecliptic based on the plane of the moon’s rotation around the earth. They divided this into 27 equal parts of about 13 degrees each, which they called nakshatra or asterisms. The names for these nakshatra first mentioned in the Atharva Veda are still used today, and are used by contemporary astrologers to calculate the dates of certain Hindu festivals.

The division into 27 days is based on the length of the sidereal month, which is actually 27.3 days. Vedic people were aware of this discrepancy and applied a periodic correction for the extra 0.3 day. So each day, the moon moved across 13 degrees of arc into a new nakshatra. They further subdivided each nakshatra into four equal pada, each measuring around 3 degrees of arc. A lot of Hindu astrology is based on the pada as unit, for example, the janampatri or horoscope that is calculated when a child is born. Each pada is associated with 3 syllables, and you pick a lucky name beginning with one of these 3 syllables based on the pada when the child was born. This is still common practice in India for the astrologically inclined. The system gets a bit more complicated because there is also a set of 9 graha or houses, and each nakshatra is also associated with a graha. Since there are 27 nakshatra and only 9 graha, the cycle of graha has to repeat 3 times each synodic month. But using this system, you can uniquely identify any period down to the nakshatra, graha and pada.

The reason I mention all this is to explain that the common or average astronomer using this system today and back in Vedic times was expected to discern a pada observationally to make any predictions at all. This meant being able to observe and identify a movement of around 3 degrees in the path of a celestial object. In contrast, modern instrument-based observations used to predict the date, hour and minute of a solar eclipse typically require measurements down to 1/20th of a minute of arc. So you can see that any predictions of solar eclipses must have been somewhat vague, predicting the year rather than timing it down to the second. Lunar eclipses were much easier, and it would have been easy to narrow them down to a day.

That’s not to say that the best astronomers of the time couldn’t be more precise than a single pada. With good eyesight and more experience, you can probably improve your accuracy and perhaps narrow down your prediction to a month or two in case of solar eclipses, but that’s just extrapolation based math and reasonable assumptions about human eyesight, the texts don’t tell us how accurate they were.

As for the mechanism of how they predicted eclipses without knowing the precise movements of earth and moon, it was based on periodicities in the orbits. One such period is called the Saros Cycle, which is the period between any given alignment of sun-earth-moon and the next repetition of the exact same alignment. It’s about 18 years and 10 days. If you observe an eclipse of the Sun, then half a Saros Cycle later you can expect an eclipse of the moon with the moon being in the exact opposite position relative to the earth.

This is easy for lunar eclipses because the earth’s shadow on the moon is huge and the eclipse can be observed from a large portion of the earth, from places thousands of miles apart. But the moon’s shadow on earth during a solar eclipse is 40 times smaller, so it’s easy to miss if you’re not located on its narrow track. A solar eclipse is also shorter than a lunar eclipse, because the earth’s diameter is bigger than the moon’s. These things make calculations harder since your data set may not be complete.

So it’s very likely that solar and lunar eclipses were being predicted in Buddha’s time, which was around 500 BC. In India, the earliest mention of making these predictions goes back at least a thousand years earlier. In China, there are legends of a couple royal astrologers being beheaded for wrongly predicting an eclipse that was supposed to accompany a royal birth. While in this particular case the astrologers made a mistake, eclipse prediction must have been some sort of business. Mesopotamia also had the astronomical knowledge and history of observations to make such predictions. The earliest Greek prediction of a solar eclipse comes right around Buddha’s time, when the philosopher Thales correctly predicted the year of a solar eclipse. The Greek literature doesn’t exactly say how he did it, but implies that he learned the technique during a trip to Egypt when he studied their records.

As for why people would do so, it was probably for religious reasons. The stars and celestial objects were significant to humans at the time, because of the widespread belief that they affected human lives.

Related: what date can be independently established for that sutra?