The thing to focus on for any kind of literature is the complete idea. Read the entire line of poetry from the first word to the end punctuation in order to get the entire gist. Reading that way will help you understand the English...it's just the syntax (the way the words are put together in the sentence) that's confusing you.
Now, Beowulf and Sir Gawain don't really give too many speeches. However, they are both heros of the people, and they have both completed larger-than-life accomplishments. Beowulf's speeches are full of boasts since he is an Anglo-Saxon creation and believes that his afterlife depends on his fame. The stories told about him after his death will provide his "forever after". Of course, he seems to have really done all the things he says he's done (killed sea monsters, swam the ocean in chainmail, ripped off Grendel's arm with his bare hands, sunk for hours to take on Grendel's mother, killed a dragon, etc.) He also puts Unferth in his place by telling Unferth and all those listening that Unferth has first, had too much to drink, second, hasn't stood up the monster Grendel and therefore that's why the creature still feasts upon his people, and third, Unferth shouldn't be speaking up anyway since he is no better than Grendel because Unferth too, killed his kinsman. In Beowulf's speech, he is very matter-of-fact and blunt about his accomplishments and what he can and will do.
Gawain, by contrast, willingly takes Arthur's place on the quest for the Green Knight. He is honorable in this respect. However, unlike Beowulf, his character is a little tarnished. He lies to the green knight about the sash--reiterating the reason for the quest in the first place. The Green Knight suggests that the knights of the Round Table have become lazy, braggarts, and less-than-honorable men since they don't all uphold the code of chivalry by which they have pledged to live. Gawain does prove his worth, however, when he asks for forgiveness and pledges to teach the rest of the knights the message.