What does grow'st from Shakespeare's sonnet 18 mean in modern English?

Shakespeare actually wrote in modern English, and the verb form you are asking about is, though now very rare, still in occasional use. Basically the story is this: if you see a familiar Word with "-st" on the end, you will probably find the pronoun "thou" lurking close by. "Thou" always takes verbs ending in -st; "thou thinkest" is the same as "you (singular) think". Shakespeare used them interchangeably. Note this passage from King Lear:

Have more than thou showest,

Speak less than thou knowest,

Lend less than thou owest,

Ride more than thou goest,

Equals show, know, owe, and go. "Growest" obviously is "grow"