The core issue is not that he didn't use the pen tool, but that there simply weren't enough pixels available to accurately represent the objects, so the edges of the objects come out looking like a (blurry) staircase. For example, the letters are vector objects, just like if he had used the pen tool to make shapes that look like letters.
To see this, suppose your canvas is only 50 pixels wide, and you want to have the letter, "O" occupy about third of the width of the canvas. This means that the letter is only around 16 pixels wide (...count them), so, of course you are going to get the staircase effect:
Now, suppose the canvas is 100 px wide and you still want the letter to be about 1/3rd of the total width. Now, the width of the staircase is half of what it was before:
Finally, if we are using a canvas that is 1000 px wide, we see a much sharper edge:
BTW, note that I intentionally turned off the pixel grid on the last two images because it would be too dense to be easily seen / counted, so don't be confused by that.
So, the question is what should the OP do in the future. The answer is very simple: If you are using PS, a program where everything is represented by discrete pixels, start with a canvas of pixel dimensions higher than you think you will ever need. One might think that if you switch to Adobe Illustrator, this issue won't arise. Well, sort of. While the shapes in that program are not built up of pixels, but are defined by mathematical curves, least this issue doesn't immediately arise. However, eventually, one has to turn the shapes in AI or from the vector tool in PS to pixels whether you are making a JPG or PNG to post on the web, or you are sending the result to a printer. So, no matter what approach you take, you eventually have to deal with pixel dimensions (as described above).
The usual next question is how do the pixel dimensions relate to quantities such as the number of "pixels per inch" (PPI) and the linear dimensions of the page in width. The answer is simple, multiply the PPI by the inches you need, and you get the pixel dimensions you need to keep your smallest features sharp.
HTH,
Tom M