How to increase size of text box without causing distortion?

When you set text along a curve the vertical lines that appear at the beginning and the end of the text are actually defining the limits of a text box. The red square with a plus sign in it in your screen shot is showing that there is text that has flowed outside of this (sort of virtual) text box. Use the black arrow selection tool to select the whole text object and then drag on the vertical line that has the red square in the direction of the curve. This will open up the "on a curve" text box and reveal the run-over text.

Your question and screen shot don't seem to agree - this seems to be type on a path rather than type in a text box. If you have text in a box, you should be able to highlight it and just increase the font size without a problem. There might be text overflow so then you would need to decide whether to increase the box size or flow the text. If you increase the box size, go to the corner handle and pull it till it all fits.

If you are typing on a path, you can also highlight the text and increase the font size. If you run out of path room, just increase the path or again you will have overflow text. You can also use a second path for the overflow.

If your question is something else, please further explain and give a screenshot with your layer panel as well.