Trade didn't necessarily stop, but its patterns were fundamentally disrupted by the eventual collapse of the Roman economy and state. /u/XenophontheAthenian has an excellent post here which looks at the importance of North African grain to Rome specifically, and its economic viability compared to other Italian grains. The Roman state could have a major impact on trade: a guaranteed grain market in Rome, say, or for the vast food requirements of the Roman military, could provide sufficient economic security for a North African farmer to invest in irrigation or a cash crop such as olives or vineyards, and in turn generate a secondary economic network to sell goods to those farmers with a guaranteed income. In turn, a state-guaranteed food supply to a city like Rome could free up members of the population to follow other economic pursuits. As a comparison, the Agricultural Revolution and the Enclosure Acts in 18th Century Britain freed up large numbers of the agricultural work force for the developing urban industrial market.
The problems come when those networks are fundamentally disrupted, whether through recession, war, plague or other population collapse. Disruptions to the North African grain supply correlate to significant population decline within Rome; by the time that supply is stabilised, the market to support it no longer exists to anywhere near the same extent, and as a result, the attendant economies - those cash crops and merchants selling to the farmers - are now no longer viable. Comparison again could be drawn to the ghost towns left in the wake of the 19th Century American gold rushes, or the continued economic struggles of Northern British industrial and mining towns or the American 'rustbelt' in the wake of the closure of the industrial facilities which provided the bulk of their employment over centuries. Like a modern day Sheffield or Detroit, the decline was a drawn-out affair and one which would require long periods of economic restructuring to arrest and then reverse.
International trade very much continued throughout the Early Medieval period, but just on a scale and on networks quite distinct from those Roman ones which had come before. In Britannia, for example, a sub-Roman population collapse likely accelerated by a catastrophic plague in the 420s as well as periodic raiding had led to a return to the smaller, more self-sufficient communities that had characterised pre-Roman Britain. Combined with the piecemeal arrival and integration of the English and the establishment of multiple polities, trade was temporarily reorientated largely internally. The exceptions were those areas with the oldest and most integrated English populations, or the closest international links. Kent, for example, maintained a lively maritime trade with Merovingian Francia throughout this period, particularly in textiles, metalwork and ceramics. As the political situation in England stabilised, so trade proliferated. The 7th and 8th Century English economy was dominated by the wics, an original network of (largely) coastal trading a productive sites which proliferated along both sides of the English channel, particularly in the Rhineland, Northern France and the East and South Coasts of England.
Trade at these sites focused largely on easily portable, industrial manufacture rather than necessarily agricultural produce. These also gave rise to the sceatta silver coinage which by the 8th Century evolved into the silver penny that was the mainstay of medieval English coinage for centuries. Perhaps as a consequence of the apparent end of industrial ceramic production in Sub-Roman Britain, the English were enthusiastic importers of Frisian and Rhineland wheel-thrown pottery until the burgeoning English wheel-thrown ceramic industry exploded in the 10th Century. Frankish trends in fashion pass into England, with straps with highly decorated end pieces starting to replace buckled belts from the 8th century onwards. The trend really takes off in Wessex with the development of the highly intricate and distinctive 'Trewhiddle' style strap ends, a fashion which then spreads into other personal wear such as brooches, clasps and seax pommels and scabbards.
By the 9th and 10th Centuries, England has a thriving international trade economy, as well as internal trade. Perhaps the most important element was the wool trade. As well as being of utmost importance for the domestic textiles market, English wool was also a major source of international revenue, primarily being exported to Flemish merchants in the Low Countries. While the raiding of the Anglo-Danish wars had severely curtailed maritime trade in the mid-9th century, by the early 10th, Hiberno-Norse settlements on the East coast of Ireland, as well as Scandinavian settlements on the Isle of Man and the Scottish Isles provided a ready market for a rapidly expanding trade network developing out of the new port cities of Bristol and Chester. Bristol in particular was a centre of the slave trade, primarily the export of Welsh slaves to Ireland, as well as the import of those slaves from Wales into wider England. The trade also focused on the import of Welsh raw silver and Irish cattle, and the export of English processed silver, metalwork and jewellery, textiles, ceramics and possibily grains.
Internally, by the 10th Century, the West Saxon fashion for 'Trewhiddle' strap ends and other personal decorative wear had spread across England, seeing particular popularity in Mercia in the early 910s, possibly as a result of Æthelflæd of Mercia or Æthelstan. Probably as a relic of Scandinavian ties to the former Danelaw areas, the style crops up in imports to Sweden and Denmark, and eventually Scandinavian 'Borre' style designs appear in England, likely both from Northumbrian productive sites as well as imported from Scandinavia.
The 10th Century Latin textbook known as Ælfric's Colloquy is formatted as a series of conversations between various community figures and a class of novices and their teacher. The Merchant in the Colloquy describes his role thus:
I embark on board ship with my wares and I sail over remote seas, sell my wares and buy precious objects that are unknown in this country. I bring these things to you over the sea enduring great danger and shipwreck with the whole of my goods hurled overboard and with me hardly escaping with my life... I bring purple cloth and silk, precious stones and gold, various sorts of clothes and dyes, wine and oil, ebony and brass, tin and brimstone, glass and like products
The mercantile trade was open to many people and organisations. We know from charter evidence, for example, that the convent of Minster-in-Thanet owned a flotilla of merchant ships that operated out of the port in London, that were excused port duties as a royal beneficence.
Much more can be said about what remains a currently debated matter, but you might find some elements of answers and introduction to the topic there. ( u/bitparity, u/Libertat, u/Tiako).