Please check out the video I've put here, it depicts what I am in need: I don't know how to resize the text box.
Only text box, not the font size.
You have to start a Text Box or Text Area, but more accurately you are creating Paragraph Text by selecting the text tool (Ctrl/Cmnd + t) and then dragging the text area out in the composition panel to create your container for the Paragraph Text.
Once you have the text area drawn out you can either paste previously copied text or type in new text. When you are finished creating the Paragraph Text you can click off the layer or change to the selection tool to move the entire text layer around or, as you have discovered, scale the text.
If you want to change the size of the Paragraph Text you have to reselect the type tool (Ctrl/Cmnd + t) and then make sure you have the text layer selected in the timeline, then click anywhere in the Paragraph text to start editing. You will see the Paragraph Text outline appear. With the text tool still selected you can hover over the corners or sides of the Paragraph Text area and change it or even move it.
All of this information is readily available as well as a bunch more info on how to handle text by simply typing Text Box or Area Text or Paragraph text or even text in the Search Help field at the top right corner of AE and checking out the community resources and help files. The first item that pops up will get you started: Creating and editing text layers in After Effects
It looks to me like you did not create Paragraph Text but just started typing with the text tool. I hope this helps. By the way, you can animate the size of the Paragraph Text area and position but hold keyframes will be applied so there will be no smooth transition from one size to another, the Paragraph area will just snap to the new position.