These animals weren't quite seen as exotic. Lions were native to Europe until around the 4th century and lived in the Transcaucasus till the 10th century. Lions were also reasonably close by in North Africa (Barbary Lions) and Egypt. As such, foreign merchants and lords often gave lions as gifts to European lords. Multiple kings are known for owning lions in their Royal Menageries, notably Charlemagne, William the Conqueror, the Holy Roman Emperor Fredrick II and the English King Henry III.

Lions were also of great significance in European culture, notably to the Romans and the Greeks, like in Herakles' 12 tasks, Aesop's fables, the zodiac sign Leo (astrology was a large part of Hellenic religion) and of course, as one of Rome's favoured animals for venatio spectacles and to eat Christians and criminals in colosseums. Later on, in Christianity, lions were also a big thing. The Tribe of Judah's animal was a lion, Mark the Evangelist was associated with lions, and the Bible makes numerous references to Lions being the boldest and strongest of all the beasts. In the 2nd century's Physiologus, a sort of Christian proto-bestiary which described the symbolism associated with animals, the lion was characterized as a royal animal. Later on in the High Medieval period, Chrétien de Troyes wrote the classic Arthurian legend Yvain, le Chevalier au Lion (if you're interested in the Medieval period, I'd recommend reading modern translations of Chrétien de Troyes' tales. He was a wonderful writer and really gives you an insight into the mentality of the time).

So, putting lions on heraldry was a way to show that your house was bold, strong, and regal. Lions in Clan Symbols (a type of Germanic proto-heraldry) were adopted around 5th century in Germanic tradition. Later on, they were adopted in a more Christian context amongst Frankish and Lombard tribes. Lions have also been present in "modern" heraldry ever since it came into appearance around the 12th century. Some of the earliest examples of Heraldry, like those of Geoffrey Plantagenet, the house of Wittelsbach, the coat of arms of Scotland, Denmark, Léon, Flanders, Swabia and Bohemia all contained lions.

So in conclusion, these animals weren't very exotic to Europeans at the time. While your medieval European may have never seen lions in person, they would've been familiar with artwork made by earlier generations, written descriptions, tales containing lions and characteristics of lions written in the Bible.

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Of all wildlife species Panthera leo (lions) achieved the greatest known geographical distribution of any terrestrial mammal.^[1] Although it should be noted that lions are probably the most-studied terrestrial species by measure of research dollar, and are in the top twenty by measure of publications about them. Lions of various subspecies have had historical distributions across almost all of Africa, existed throughout North America, parts of Central and northern South America, and substantial parts of Eurasia except for "southeastern forests."^[1] This includes parts of what is today the United Kingdom.

This [map] provides an overly simplified view and includes areas that obviously would have been uninhabitable by lions (the Sahara, for example, as well as all of Alaska). This distribution map depicts more recent occurrence, from 1750s onwards, for comparison.

It's believed that the lion of North America, Pantherea leo atrox, likely were hunted to extinction during the Pleistocene (the most recent Ice Age ending about 11,500 years ago)^[1] but had lived alongside dire wolves and horses.^[2] The European cave lion, Panthera leo speclaea, were 25% larger than today's largest African specimens (up to 250kg or 550 pounds) and appeared in Europe about 600,000 years ago. Cave paintings dating back 200,000 years suggest P. leo speclaea existed at the time of early humans,^[2] so lions in a general sense have a strong basis in human culture from very early times.

The range of Panthera leo throughout Eurasia similarly has been curtailed by human expansion. Although a small population still natively occurs in India, and is considered the Asiatic lion, this population is likely the remnants of migrations of Panthera leo leo (the African subspecies), eastward across the Middle East.^[2] In fact, Sri Lanka's Sinhalese ethnic group derives its name from "lion race" and cultural depictions in art and folklore suggest that lions as a cultural symbol, if not as a species, existed on the island prior to the species largely dying out in India. And while eastern China is not believed to have had present any populations of lions at any time, they similarly developed lions into important icons and symbols of power and would have come into contact with live lions likely by importing them from India around 1,000 years ago.^[2]

Although lions no longer exist in the Middle East, small and isolated populations of leopards and cheetah still exist in Pakistan and Iran, with cheetah no longer extant in India though it has healthy populations leopard. All three species have been featured in important cultural traditions in those regions and, particularly the lion, on symbols of royalty and later on some national flags. All three species are considered national symbols of Iran, with the lion featuring prominently in medieval Persia in the 12th century and probably adopting much of the symbolism derived from other major cultures it came into contact with during that time.^[2]

During and after the Crusades lions became popular in heraldry in European nations, though it's not immediately clear to me whether there were wild populations extant in the Middle East at this time, it is known that as far back as 1125 King Henry I had lions and leopard in captivity for viewing and possibly for hunting.^[3] So lions (P. leo leo in this case) would have been very accessible to certain nobility as a result of trade. Certainly during Biblical times P. leo leo existed in notable parts of the Middle East, including Palestine, and so stories in the Old Testament would have drawn on actual occurrence of lions.^[2]

Ancient Egypt had a long history of reverence of the lion, as well as other species which no longer natively occur in modern Egypt, like the hippopotamus. Roughly 1,000 years ago Bastet, originally a lion-headed goddess, emerged as a popular diety and a focus of cat cults (if that's the appropriate term). Bestet's sister Sekhmet was also depicted as lion-headed and was associated with the sun's destructive abilities or powers and associated with "wrath and vengeance." From these associations it seems that lions became associated with the sun god (Osiris, the father of Sekhmet according to some myths) and also with cats which became important symbols and were an element of worship in Ancient Egyptian culture.^[2]

Ancient Rome also had a history with lions, although due to the breadth of this I will leave that to someone else to tackle. And I believe animal usage in amphitheaters and other venues have been discussed on /r/AskHistorians multiple times.

As to the prevalence and import of these species on coats of arms and other heraldry, I'm sure someone will come along to provide insight into these relationships and symbolisms.

Sources:

[1] Walker's Mammals of the World, 6th Ed., Vol. 1, Ronald M. Nowak

[2] Wild Cats of the World, Mel Sunquist and Fiona Sunquist

[3] The Ark in the Park: The Zoo in the Nineteenth Century, by Wilfred Blunt