The one-word answer: seeds.

Most of the important Columbian Exchange plants were food crops with long histories of cultivation in either the New or Old World. One important characteristic of many popular food crops is that they are easy to grow; this is especially important for annual crops, where the plants only live for one growing season. Some important food plants, and some important non-food plants, which were not grown as crops, took much, much longer to move across the Atlantic or Pacific, or never did. Examples include the Brazil nut (the tree can grow elsewhere, but only a few types of bees are suitable for pollination, so even today production depends on wild trees) and the rubber tree Hevea brasiliensis.

With many of the crops, the seeds themselves are the part that is eaten, and all that is necessary is to reserve a part of the harvest for planting for the next growing season. Important examples include maize, wheat, beans, peas, and peanuts. With some other crops (chillies, tomatoes, pumpkins), it is the fruit that is harvested, and some seeds need to be saved from the fruit for future planting. With these crops, it is often sufficient to just transport the fruit (possibly dried, e.g., as common with chillies).

There are other crops grown from seed where seeds are not included in the harvested part. Production of seed often doesn't coincide with harvesting, and collection of seed for future planting must be done separately. These plants can still be transported as seed, but more local knowledge and/or cooperation is needed. For example, one cannot grow tea or tobacco from imported tea or tobacco leaves.

Some other crops, while they produce seeds, are more usually grown from cuttings of stems or tubers. Stems and tubers for cuttings, while alive, generally have low energy demands and plenty of stored nutrition, they don't need light or water during a not-too-long voyage. Where the part used for cuttings is the part harvested as the crop, it is sufficient to transport the harvested crop. If it is a different part of the plant, then some local knowledge and/or cooperation is again needed. Examples include sugarcane, potato, sweet potato, cassava, Old World and New World taro.

The important Columbian Exchange plants were those that could easily travel as seeds or cuttings, and were easy to grow from seeds and/or cuttings. Not only did this enable them to make the crossing in the first place, but also helped them spread around Africa and Eurasia or the Americas after making the crossing.

The Columbian Exchange was also facilitated by the wide range of climates on both sides of the Atlantic and Pacific, from wet to dry, from tropical to temperature. Tropical Brazilian crops could find suitable homes in the Old World (e.g., cassava, one of the most important of the Columbian Exchange crops), and North American crops in Europe.