Hmmm. Difficult question, but I'll give it a stab with three suggestions...
1.
Foreshadowing. If you're running a lengthy campaign, establish early in the piece that there's this incredibly powerful organization or entity. For bonus points, don't make it obvious that it's going to be the Big Bad. For example, at 1st level, start the party out in a cathedral dedicated to a good god. They're there to pick up a courier delivery. While they're waiting in the foyer, the high priest is seen kneeling before the alter and a huge angelic creature (solar angel) is speaking in Celestial to him, issuing the edicts of the faith. Choirs of angels wait humbly in the wings, and golden shafts of light stream down through the stained glass windows above. The PCs have no real interaction with this scene, but it establishes the church and the high priest are heavy hitters. At 3rd level, while they're adventuring, one of them receives a dream from that very solar angel - the PC has an important destiny! Wow; they're now being addressed by an insanely powerful creature directly. At 5th level, angelic servants guide them to the location of a powerful artifact of the faith. At 7th level, they start treating with the high priest on an equal basis. At 9th level... SURPRISE! The solar is a fallen angel bent on leading the church on a fanatical crusade to wipe out all heretics (which includes other good-aligned creatures that aren't part of the faith). The heroes are now squaring off against the Legions of Heaven...
Anyway, you get the idea. Find ways to blow up the party's expectations of level-appropriate behavior. Don't start with kobolds and small villages at 1st level. Have them running from Alduin the World-eater Dragon as their introduction to the campaign, or give them the One Ring while they're attending a party in the Shire. Make the party feel like they're part of something way bigger than they are, even if they can only interact with it in limited ways... at first.
2.
Props. The obvious one here is mass battles, but this is budget-dependent. If you have a box full of Dwarven Forge 3D dungeons, or some painted Warhammer terrain, and a couple hundred orc minis... then you can do something fairly jaw-dropping. But there are other ways of achieving the same feeling on a budget. Here are some random ideas: background music or sound effects (sounds cheesy, but it actually works - plenty of material available for free on the web), invest in a single 28mm scale ship model and some blue fabric (for a nautical campaign), get a few pouches of old coins and hand them out when PCs find treasure (e.g.
https://www.kickstarter.com/projects/drawlab/legendary-metal-coins-for-gaming/description), fake up an old treasure map (
http://www.hexographer.com/ or
http://www.dungeonographer.com/) by applying a filter and ripping/staining the print-out, etc, etc.
3.
History. This one takes time (heh), so I don't know if it's suitable for your group. My own campaign world has been running 30 years in real-time (yeah, I'm old). However, that's not a single campaign. That's a bunch of campaigns... and every now and then, we advance the timeline of the world. Nearly 300 years have passed in game-time since the first session, which means you have characters today who are descendents of characters from 10 years back (real-time). Successful previous characters are rulers of nations or other major entities, and the players get a sense of both the passage of time and their impact on the world. You can achieve the same thing in a single campaign by making it clear that time is advancing, especially through the use of Downtime rules.
This is an example from one of my current campaigns. The party did typical adventures until 3rd level, then had 4 months of downtime while their employer didn't have any work for them. There isn't a constant stream of dungeons and perils, after all. At 5th level, they ended up shipwrecked on an island and spent 3 years (!) making their own ship (!!) to get off the island. By the time they got back, one of their romantic interests had moved on to someone else, and one of their business interests had economically collapsed. This created a whole new set of story challenges. At 7th level, they decided to take on some pirates - but I made it clear the pirates couldn't be beaten by simple "adventuring". They needed to gather a fleet, which took gold and time. 5 months later, they're still pulling together the necessary resources.
This is basically a direct answer to the old CRPG problem of going from level 1 to level 50 in about 7 days. In games like Skyrim or World of Warcraft, for example, you can basically become an archmage or noble in about a week of adventuring. But in tabletop RPGs, the DM can control how much time passes. By spreading a character's career over years, and having the background world change at the same time (borders shift, NPCs age and marry and die, faiths arise and fall, etc), you can create a certain sense of scope and grandeur.
Anyway, just my two cents. Good luck.