What's he saying?
"Since I left you, mine eye is in my mind; / And that which governs me to go about"
Since I left you, my eye no longer functions as it should, but rather sees only what my mind wants it to; so my eye, which helps me do most activities
"Doth part his function and is partly blind, / Seems seeing, but effectually is out;"
can no longer do its normal job, though it seems to, since its vision is blotted by my mind;
"For it no form delivers to the heart / Of bird, of flower, or shape which it doth latch:"
My eye can't report back what it actually sees
"Of his quick objects hath the mind no part, / Nor his own vision holds what it doth catch;"
The mind does not share the living objects seen by the eye, and the eye cannot hold onto what it sees;
"For if it see the rud'st or gentlest sight, / The most sweet favour or deformed'st creature,"
Regardless of whether the eye sees beautiful things or ugly things,
"The mountain or the sea, the day or night, / The crow, or dove, it shapes them to your feature."
If it sees mountain, sea, day, night, crow, or dove, it really only sees you.
"Incapable of more, replete with you, / My most true mind thus maketh mine eye untrue."
The eye is unable to see anything else, because it is so filled with you; thus does my faithful mind make my eye incorrect in what it sees.
Why is he saying it?
Sonnet 113 is a continuation of the previous sonnet, addressing how totally the poet is consumed with images of the fair lord. In Sonnet 112, he declared that nothing else in the world has any effect on him whatsover, claiming in the final couplet, "You are so strongly in my purpose bred / That all the world besides methinks are dead." Now, in Sonnet 113, he is so taken by his beloved youth that he no longer sees his actual surroundings. Instead of reporting back to the mind images of "The mountain or the sea, the day or night, / The crow, or dove," the eye transforms everything it takes in into an image of the fair lord.
This sonnet begins with the phrase, "Since I left you," implying that the separation is the fault of the poet himself. It is unknown how long the period of absence lasted, or its reason. It is possible that it refers to the absence of Sonnets 50 and 51, in which he complained about being borne farther and farther away from the fair lord on horseback, or that of Sonnets 97 and 98, in which he compared the distance from his beloved to the barren winter, while being in the presence of the fair lord was like the spring. Or, it could refer to the emotional distance caused by the poet's philandering, referenced in Sonnets 109 and 110.
The relationship between the eye and the heart was dealt with in previous sonnets. For example, in Sonnet 24, the eyes are personified as painters, recording the image of the fair lord in the speaker's mind without input from the heart: "Yet eyes this cunning want to grace their art; / They draw but what they see, know not the heart." In Sonnet 46, they are "at a mortal war" over how to divide the experience of the fair lord's presence, and they finally decide that, "...mine eye's due is thy outward part, / And my heart's right thy inward love of heart." In this sonnet, it is unclear who is in charge: the eye or the heart. It seems that the eye works independently of the heart in lines 8-12; no matter what it actually sees, "it shapes them to your feature," reporting back only images of the fair lord. However, the final couplet implies that the eye is functioning this way because it is being commanded to do so by the heart. Line 13, "Incapable of more, replete with you," refers to the heart, which is totally obsessed with the fair lord. It is "most true," meaning faithful, but makes the eye "untrue," or false in its observations.
The word "form" in line 5, "For it no form delivers to the heart," can be read as referring to Plato's idea of ideal Forms. In this theory, Forms are the most fundamental and highest kind of reality; the material world we exist in is only known to us through sensation, and cannot provide genuine knowledge like the Forms. As is declared in many sonnets, the fair lord is the ideal of all beautiful things in the world. Thus it makes sense that the eye would choose to focus on the image of the fair lord rather than the material world upon which it actually looks.
There was no scientific understanding of optics at this time; theories of how sight worked were based on ancient ideas. The eye was thought to function by sending out a flux that latched onto the object that was being seen. In this sonnet, the eye is incapable of accurately representing to the mind any "shape which it doth latch;" since it cannot "hold[s] what it doth catch," or retain the actual images upon which it looks. In this context, the terms "heart" and "mind" are interchangeable as interpreters of the eye's report.