In the years leading up to The Hobbit, thousands of Dwarves were desperate to reclaim their home, yet most believed it to be a lost cause. So much so, that only 13 Dwarves went on the quest to Erebor, along with Bilbo Baggins and Gandalf. While Thorin Oakenshield led the charge, it was actually Gandalf who encouraged the adventure, and he definitely stood out amongst the others. So why did the wizard join the company of Thorin Oakenshield?
More so than any other wizard, Gandalf enjoyed traveling Middle-earth and aiding those in need. But with his level of power, he mainly focused on the most important tasks of the world, more specifically, keeping the darkness at bay. Each wizard was sent to protect the world from Sauron, and for thousands of years leading up to The Lord of the Rings, that's exactly what Gandalf did. So with this in mind, it gives a hint towards what Gandalf was really doing with Thorin's team.
When Tolkien wrote The Hobbit, much of the lore from The Lord of the Rings was unknown, and he hadn't even planned for them to share the same world. And so, the original reason for Gandalf's involvement was simply that he was a jolly old wizard who wanted to help. But afterwards, the book Unfinished Tales of Numenor and Middle-earth was released. Within this, some extra scenes missing from The Hobbit are included, one of which explains Gandalf's true intentions.
During a conversation between Gandalf and Thorin, the wizard says, "Far away from here, I wonder if you fully realize the strength of a great dragon. But that is not all: there is a Shadow growing fast in the world far more terrible. They will help one another." And so, with Sauron's ever-growing power, Gandalf was worried he would reach Smaug first and cut a deal so that they would work together. He knew if Sauron was able to get a dragon on his side, the coming war would be lost far more quickly.
Gandalf also explains, "To resist any force that Sauron might send to regain the northern passes in the mountains and the old lands in Angmar, there were only the Dwarves of the Iron Hills." So in other words, Gandalf knew that once Sauron's invasion began, northern Middle-earth wouldn't be strong enough to defend itself. Meaning the Lonely Mountain needed to be armed with Dwarves to help fight the war.
Even during The Hobbit's quest, Gandalf had to leave before reaching Erebor. It's later revealed that he had to deal with a "Necromancer" in an old fortress, which eventually turned out to be Sauron. So while Gandalf did want to help the Dwarves reclaim their home, the real motive was always to counter Sauron. And in the end it paid off, as during The Lord of the Rings, the massive Easterling armies were locked in a stalemate against Erebor, and failed to take the Dwarven kingdom.