"To see to something" means "Taking care of," or, in this case "Making sure it will happen." Like:
"I will see to your needs." -> I will take care of your needs.
In your sentence, you need an 'it' (as object) to bridge the phrase with 'that': "I'll see to it, that..." Theoretically, you could say: "I'll see to your wishes being carried out." But that's very ugly! "I'll see to it, that..." is much, much better, and far more common.
"I'll see to it (that) your wishes are carried out." -> I will make sure your wishes are carried out.
"To see to" carries the meaning over 'overseeing' the process (hence, has the added meaning of 'making sure it will happen.').