Translation:Let me check the words on that page.
To create the "let me" meaning. -u ending signifies imperative (do, come, check..).
There's no imperative for the first person in English, or is there? To command oneself to do something doesn't make much sense. Does it?
You are right. I don't know much about the English grammar, but in some some languages it is called the volitive. In esperanto -u ending is used to indicate a command when used for non-first-person people, but it becomes the indicator of volitive mood when used for I and we.
You can certainly say to yourself "Just do it!" or "Jump!" Exactly as one would order others.
The voice speaks a bit too quickly here. It's hard to tell what's being said.