What does this sentence, "wish can stay longer yet too busy", mean in simple English? What phrase does it use? The "can...yet" confuses me.
First of all, the sentence is ungrammatical as it's written.
In the original, wish can means I wish I could (or I wish I were able to), and yet is a conjunction similar to however, but, or still.
To both rephrase it, and better convey its meaning:
I wish I could stay longer; however, I'm too busy with other things, and I need to leave so that I can take care of them.
It sounds like something that someone texted you. In texts it's common to leave out words, as long as the result can still be understood. If written in normal, grammatical English it would be,
I wish I can stay longer but I am too busy.
That being said, the use of "yet" in your example is odd. While "but" is a simple conjunction that indicates the following statement will be the opposite of the previous statement, "yet" implies a kind of a continuation of the statement in a different direction. Examples:
I didn't want to lend him the money, yet since it's what a good friend would do, I gave it to him.
The sun hadn't set, yet we went to bed, exhausted after the long day.
In this case, it's either a trivial mistake, or else it implies much more than would be expected in this situation, for example:
I wish I can stay longer, yet (since we don't always get what we want and I am) too busy (I can't).