This sonnet is a fairly straightforward statement of complaints – that is, it is not an elaborate “conceit” but rather a real lament in natural "realistic" language (although  in iambic pentameter) of the narrator’s impoverished state, both financially and socially (“I all alone beweep my outcast state”), a “pity party” for eight lines, then a reversal “marked by “Yet” for four lines, in which the narrator remarks that his melancholy changes, ending with the resolving couplet (“For thy sweet love rememb’red…”) – a perfect example of the Shakespearean sonnet form.  Aside from one obvious simile – “Like to the lark at break of day arising” --  (“wishing me like…” is not a simile, but merely a statement of a wish), the figures of speech are restricted to hyperbole (that is, exaggeration; for example, “I scorn to change my state with kings.”) and a little personification (“When in disgrace with Fortune and men’s eyes…”).  “Sings hymns at heaven’s gate” might be seen as synecdoche, where one detail, “gate,” stands for the whole (heaven).  One of the reasons this sonnet is so popular is because it is transparent, not clouded by figures of speech. (As for "why," that is unanswerable.)