The most obvious answer is that they did not, the bulk of the trade between Rome and India was indeed done through the ports on Egypt's Red Sea--the so-called "maritime Silk Road". Maritime trade around the Arabian Peninsula has a long history, the earliest clear marker being pearls during the Neolithic, and famous early examples include the pharaohs Sneferu and Hatshepsut's expeditions to "Punt"--probably Somalia (worth noting these certainly did not constitute the sum total of exchange between pharaonic Egypt and the Red Sea, they are just well publicized examples). Ptolemaic Egypt began constructing ports on the Red Sea to facilitate the transport of elephants to be used in their armies, which was an important part of early hellenistic warfare, and the direct, monsoon route between the Red Sea and the Indian Ocean was traditionally believed to have been discovered by the Ptolemaic sea captain Eudoxius, or perhaps Hippalus. During the Roman empire, this commerce expanded considerably, to the point that the Roman state derived considerable profit from tolls on the route, and the "Yavanas" (Greeks) bearing gold and wine are recording in the sangam poetry of south India. This commerce saw its pea around the middle of the first century CE, and experienced a sharp decline at the beginning of the second century, and nearly disappeared during the third century before experiencing a revival in Late Antiquity.

This paragraph leaves out a lot, it smooths over a lot of complications, it glosses over a lot of contentious scholarly debate, and I am happy to expand on any part of it, but for now we will call it "good enough".

To actually think about this is terms of goods, however, we have three key ones: Roman gold in the form of coins, Roman wine primarily from Italy, and black pepper from India. These can be considered the "bulk items" that provided the engine of trade, but there were a whole variety of other goods, recorded ones being parrots, ivory, cinnamon (discovered in Pompeii!), rice, weapons, and, of course, silk. Silk is its own complicated thing, but for now let's just say that it came from China, and it traveled through a number of different ways, the most important ones being through the Taklamakan, across the Himalayas, and into India (this is the route of the pilgrim Xuanzong as immortalized in Journey to the West), or down through Yunnan, into the Bay of Bengal, and either around the coast of India or overland. What is interesting in this context is that the great archaeological site of the "Indo-Roman trade" is at Arikamedu, on the East side of the subcontinent, but this is still a somewhat poorly understood region.

This is not to say that overland routes were entirely unimportant, one that gained increasing prominence probably in the second century is through the Persian Gulf, into Mesopotamia, and across to the Mediterranean via the great city of Palmyra. But during the height of the trade, overland routes were very much secondary, and Roman writers consistently thought of India as "across the sea" rather than across the land".

There is a ton I am leaving out, including the very Silk Road itself (this is the name given to commerce from China into Central Asia in the early twentieth century and only later was the name for the entire Marco Polo journey) but this is a basic overview. Consider this something of a jumping ff point from which I would be happy to answer further questions.

I think Rome and the Distant East by Roaul MacLaughlin s a best easily accessible work on the topic, although I have a number of gripes with parts of it. Roberta Tomber's Indo-Roman Trade: From Pots To Pepper is a much more detailed archaeological examination of the topic. Rome and the Indian Ocean Trade from Augustus to the Early Third Century CE by Matthew Cobb is a book I have not gone through, but is rather more recent than those and from what I have read is very good.