O who shall, from this dungeon, raise A soul enslav'd so many ways? (1-2)
For the soul, life is like an unending D&D session. It's trapped in a dungeon and forced to move and sense and operate according to the whims of this hulking skeleton.
With bolts of bones, that fetter'd stands In feet, and manacled in hands; (3-4)
The main image here is of a soul completely immobilized inside its body. Like a criminal, its freedom is compromised with a material prison: handcuffs, manacles, and bone-rods.
A soul hung up, as 'twere, in chains Of nerves, and arteries, and veins; (7-8)
But if you thought being cuffed up was bad, get a load of what else this soul endures. By describing itself as "hung up" in chains of nerves and veins, the soul deliberately compares itself to a torture victim. The take-home here is: get me outta here!
O who shall me deliver whole From bonds of this tyrannic soul? (11-12)
The soul complains more about its lack of freedom, but the body gets a little political rhetoric of its own in at the beginning of its first stanza. Plus, the soul was a little more coy about everything, complaining through metaphor and paradox about all the sad bodily stuff it endures. The body, on the other hand, comes right out and says it: the soul is a tyrant and it's keeping me in slavery.
What magic could me thus confine Within another's grief to pine? (21-22)
It's interesting that the soul talks about "magic" instead of God or some other religious force. Maybe it's so disgusted with the reality of interacting with a body—pain! flu! sweat!—that it can't imagine anything spiritual locking it up in this madhouse. It must be some kind of sorcery.
Constrain'd not only to endure Diseases, but, what's worse, the cure (27-28)
And there's a touch of irony for ya. Because the soul would be happier without the body, it's not so secretly hoping for the grim reaper to come knocking. That's why diseases are bad but cures are worse: this opinionated amorphous blob can only get its freedom through death.