Other metal subgenres have their own common theme (eg. thrash metal has politics, black metal has satanism/occult, brutal death metal has gore, doom metal has depression/drugs). For power metal, many bands have a common fantasy theme involving dragons, magic, warriors, anything D&D/Tolkien/fairy-tale related. There are non-fantasy power metal bands such as Sabaton (history) and Sonata Arctica (emotions) (although the former is more associated with heavy metal), but many bands like Blind Guardian, Rhapsody of Fire, Nightwish, and Hammerfall tend to be like this. I know that escapism and spectacular imagery/lyrics as a whole is a general thing in metal, but how did fantasy emerge in power metal specifically? Did bands like Dio or Manowar play a role in this? Maybe it's because power metal is more "lighthearted," so they go for more friendly themes/imagery revolving the beauty of the world.

Dio.

Plus metal has a good fit with fantastic lyrical imagery.

It is interesting that two of the earliest Euro power metal bands to leave a big impact (Helloween and Stratovarius) didn't have the fantasy themes though. Okay yeah so Keeper of the Seven Keys has the imagery on the cover and in the title track but otherwise it's mostly real life based stuff. I think Blind Guardian may have been one of the first power metal bands to fully embrace fantasy and reach a wide audience. But yeah, like Ace_Rimmer said a lot of it comes down to much of the genre consisting of a bunch of Dio fanboys lol.

Dio and Rainbow were the major musical influences on that melodic sound, no question.

The other major influence was film, and the inspiration for these movies, such as Robert E. Howard's The Shadow Kingdom. The 1980's saw a huge increase in sword and sorcery films, which were hugely popular, Arnold Schwarzenegger's Conan the Barbarian being the most popular of them. The imagery from these films meshed well with power metal, and was an obvious source of inspiration.

only those graced by encountering the magical unicorn in the woods are blessed with the ability to compose pure cheese.

The roots for it can be traced back to the influences of Dio, Rainbow, Manowar, Fates Warning and a few other bands in the early to mid 80s that had a particular fascination with the old Sci-Fi, pulp fantasy and high fantasy literature of the late 19th and 20th century, but I'd actually argue that the phenomenon itself didn't really become codified until you get to the 2nd wave of European power metal, specifically the ones who began cropping up in the mid-90s such as Nightwish, Rhapsody, Nocturnal Rites, Kamelot and Hammerfall. I'd say the only band that could be treated as a lyrical forerunner to this phenomenon in the first wave (i.e. the stuff that occurred between the mid-80s and the early 90s) is Blind Guardian, as noted by Metallic Shock, and musically their signature sound was a bit heavier and more speed metal-oriented than what became the norm in the mid to late 90s. If you really parse the stuff that was flying under the radar between the release of Helloween's Keepers albums and when Nocturnal Rites' In A Time Of Blood And Fire and Rhapsody's Legendary Tales hit the scene, the lyrical content was very different. Even bands that sounded really close to the European sound in question during the early 90s like Abraxas and Chroming Rose were more focused on historical events and real life situations.

Hells_unicorn offered a very insightful and complete post. I guess if we consider not only Dio and Rainbow, other early lyrical influences would be Coverdale-era Deep Purple (Mark III and IV) and, then, in the early-1980s, underrated and unappreciated British bands such as Angel Witch and Demon - those bands showed the connection between occult and fantasy, which Dio also explored or at least often hinted at. Demon's lyrics are masterful. Incidentally, that band was covered by Blind Guardian, so they were an influence at least on some people.

This is the answer to most things in life, actually.

its my standard answer to most questions and is usually appropriate.

Fantasy-based lyrics have had a seat at the table in metal all but from the get go, even prior to it becoming a full distinct genre, you had certain forerunner bands like Led Zeppelin writing music inspired by Tolkien's books. You can find individual songs from almost every American and British band in the early 80s dabbling in fantasy lyrics, it just wasn't common to get a full album worth of it until you get to the 90s. If you parse all of Dio's solo albums prior to Magica, he devotes roughly half of the album to fantasy, the other half is usually based in reality, and Dio wasn't above putting out a crowd-pleasing banger with lyrics praising his audience (see "Stand Up And Shout", "We Rock", "Here's To You", etc.)

In all honesty, the whole notion of writing an entire conceptual metal album based in high fantasy alone doesn't really become a thing until Rhapsody's Legendary Tales in 1997.

Progressive rock plays a part.

Many bands like ELP, Blue Oyster Cult, Hawkwind, Yes, Jethro Tull (etc) had a science fiction/fantasy bent to their lyrics, to varying degrees. When you listen to something like "Karn Evil 9" (with its futuristic scifi lyrics, spoken-word sections, immense length, ridiculous guitar/keyboard soloing, Bach inspired sections) you've got a lot of the raw elements of 90s power metal.

Yeah, 70s prog and heavy rock (add Rush, Uriah Heep, etc) and Dio really. I'm pretty sure the "nerdy" culture was huge among metalheads (D&D, Tolkien, films like Excalibur or Krull, etc) and you see that in the lyrics too.

Powerful and epic music ----> A need for lyrics/aesthetic to match the music ----> A tendency to delve into fantasy, mythology and history.