1. The 着 in 我戴着帽子是因为我刚刚去剪头发 extends the action of the verb 戴. So adding it changes “wear” to “wearing”.

2. Compare the following two sentences:

a. 我戴着帽子因为我刚刚去剪头发

b. 我戴着帽子是因为我刚刚去剪头发

The second sentence adds a 是 because the speaker wants to emphasize that he is wearing the hat solely because of his new haircut. In English, inflections in tone might be used to express such emphasis.

3. Notice how 好 can be used to mean “very” in place of 很 in 看起来好酷 and 可是我觉得这个发型好丑.

4. In 我才发现, the 才 emphasizes that the speaker found out later than he would have liked to.

5. Notice the construction of the sentence 那帮你剪头发的那个人他犯了什么错误? In Chinese it’s common to see the subject described first, followed by the action. So the literal meaning here is “So the person who cut your hair, what mistake did he make?”

You can see a similar construction in:

后面的，可以剪短一点没关系.

听这个女人说话的口气，这个男人应该不常戴帽子.