What does "carry't" mean?

In Early Modern English, vowels that didn't represent any sound were often dropped and people indicated this omission by using an apostrophe.

It was mostly used in past tense and past participles when the extra syllable from the -ed endings¹ was dropped, so you'll see that in Middle English texts, the omission of a vowel is often indicated by an apostrophe. Sinned would've been written sin'd, loved would've been lov'd, brushed would've been brush'd etc.

Shakespeare's use of the apostrophe is marking where certain letters shouldn't be pronounced.

So it's carry it, the apostrophe indicates the omission of the i.

According to A Complete Concordance Or Verbal Index to Works, Phrases and Passages in the Dramatic Works of Shakespeare:

Carry it. He will carry't, he will carry't; 'tis in his buttons; he will carry't

He's used it for carry it in other places too.

Shakespeare has also used apostrophes for other contractions such as 'tis, 'twill and 'twas and pronunciations such as 'cause, strok'st, and o'er.

This blog gives another example where apostrophes have been used to indicate the omission of vowels in certain positions:

Christians d'obey th'officers and rulers, that b'appointed of God in th'Earth
[LinguistList]

As explained in The History of Early English by Keith Johnson, apostrophes were used to replace unstressed syllables in -ed endings, lightly pronounced medial vowels, letters in prepositions and pronouns etc.

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^{1: the past tense marker -ed used to add a separate syllable to a word back in Middle English. In Early Modern English, however, the extra syllable was lost probably because it was unstressed. At some point, the e from -ed was removed and people started using an apostrophe to indicate its omission in spelling.}