The speaker grieves, or bemoans his existence because he doesn't have the money, art or scope he sees in other men. He uses phrases like "rich in hope" which means promise, in particular reference to financial hope, but hope for prosperity in general. The speaker, wallowing in his "outcast state," is jealous; he speaks of his wish to be "featured" like a man with many friends or some art that makes him famous. The speaker is essentially an 'everyman' wondering what it's like to be a king or someone famous, rich, etc.
At the height of his self-loathing, the speaker thinks on "thee," his beloved and his spirit soars like a lark. This is where the turn takes place. "Lark" implies soaring away from the sullen Earth, to heaven and also, 'lark" as a verb means to play. So this love he's just remembered is more important than the wealth or life of a king. To be clear, it is the speaker's memory of "thee" which brings him out of his self-deprecating state. In short, this sonnet is about material wealth and social standing versus romantic love and the ability to mentally choose, via memory, and to embrace the 'everyman' life over the life of kings
The speaker is discussing all of the bad things that have happened to him in his life, and how thinking about his love makes him feel like life is not so bad.
Some of the specific things he mentions are: when others look upon me disgracefully, when I am alone, when I cry to heaven and feel unanswered, when I am envious of others, when I feel no pleasure from things I once enjoyed, when I despise myself
The turn takes place in line 9, when the speaker transitions into happier thoughts from previous thoughts of self hatred and depression.
"Yet in these thoughts myself almost despising,
Haply I think on thee, and then my state
Like to the lark at break of day arising
From sullen earth sings hymns at heaven's gate;"
It is usually easy to identify the turn in a sonnet because Shakespeare will include a transition word like "Yet" or "Though" somewhere around line 9. This indicates a change in subject matter or mood.