I'm not aware of a "system of equations palette", so I may be missing something, but I believe what is happening is that you are using a form from one of the palettes that is intended for styling and layout of stuff. By using it in an Input cell, Mathematica is expecting to be able to evaluate whatever you put there. I don't know what that left bracket thing is supposed to mean for evaluation, so I don't know specifically how to explain its use.

One thing I can point out is that you are using Set instead of Equal. If this thing really is supposed to solve a system of equations, then I suspect it expects Equal.

But if you really just want to solve a system of equations, you could use Solve:

Solve[{x + y == 1 && y == 2}, {x, y}]
(*{{x -> -1, y -> 2}}*)

If your objective is to solve a system of equations, use Solve. If your objective is to typeset something that doesn't need to be evaluated, put it in a text cell (or any cell type you want that does not have the Evaluatable property).

Maybe there are folks with more palette expertise, but until they chime in, all I can tell you is that I've searched all of the standard palettes and I can find no evidence that there is a typeset version of Solve. There is a placeholder version of DSolve:

There is also a form that looks similar to the one you're using, but it's associated to Piecewise:

Notice how it has 4 placeholders. Also, hovering over it shows you what it's for.

For what it's worth, you can create your own palettes. So, you may have to resort to building a custom palette.