This is one of Neil Young's strangest, most surreal songs. Despite a few decades of scrutiny by Young fans and critics, it continues to elude all attempts at rationalization. It's likely there's no concrete underlying meaning at all, as Young himself is quoted in Shakey as saying he had no intended meaning for it to begin.
"Some do, some don't," Young said in regards to whether or not his songs have a definitive meaning behind them (specifically mentioning "Last Trip to Tulsa" in the discussion). "It's not important to me."
In the song, Young seems to be recounting past lives or identities. He identifies himself as having been a cab driver, a woman, and a folk singer. He also mentions having once been asleep, driving down the freeway, and chopping down a palm tree. There are frequent mentions of cars, roads, and travels.
The most popular interpretation among fans has been that the song is about drugs - specifically LSD. There are mentions of yellow servicemen, green gasoline, Indians trying on Young's clothes, and other assorted oddities, so it's not a big reach to connect the two. Still, the usual "it's drugs" line doesn't seem to fit Young's modus operandi. Though he's been open about his own drug use, he's also never been one to glorify it. While he certainly may have dreamed the song up while stoned, it seems a little too obvious and unimaginative to have the song actually being about some drug experience.
The song starts with discussions of driving a cab and then ends with him dropping a palm tree down on a friend's back as the man walks to his Cadillac. This occurs after Young's admission that he's been cutting at the tree for 87 years.
Young turns a lot of phrases that seem like they have some deeper meaning, such as:
So I unlocked your mind, you know
To see what I could see
If you guarantee the postage
I'll mail you back the key
Despite these gems, the song as a whole simply doesn't give itself up to any kind of cohesive interpretation.
By all appearances, and by Young's own account, any deeper meaning arose from the darker corners of Young's mind, and not from any purposeful meaning. It's a stream-of-consciousness song. This fact can be maddening to fans who want a concrete story. On the plus side, though, listeners are free to shape the song in any way they choose.