It's funny to see a hatchling like you beaten by the old one. (16.35)
Very funny, Saphira. Actually, it might be funny if we got to see young Eragon get absolutely trucked by old rickety Brom. Here Eragon learns a valuable lesson from his elder sparring partner: strength comes from experience.
The energy inside him burned at an unbearable level. […] the air resounded with an explosion. A blue shockwave blasted out of the monster's head, killing the other Urgal instantly. (18.44)
Whoa. Here we witness the raw power of Eragon's magic for the first time. Of course, it takes a lot out of him, but we'd rather be drained than have our heads exploded by a supernatural bolt of force, wouldn't you? This is the first hint of the kind of awesome power that Eragon is capable of.
"This magic—for it is magic—has rules like the rest of the world. If you break the rules, the penalty is death, without exception." (19.44)
With great power comes great risk. Brom lets Eragon know in no uncertain terms that his magic skills will make him pay if he doesn't learn to control them. Isn't that a good thing, though? How might Eragon's use of magic be different if he could just blast away at things without any consequences?
"Magic takes just as much energy as if you used your arms and back. That is why you felt tired after destroying the Urgals." (19.65)
You wouldn't think that a supernatural skill would take such a physical toll on a person. In other words, Eragon can't simply wave a wand around, a la Harry Potter. His power comes directly from his own physical being. It takes something out of him every time he uses it. In that way, his magical powers stem directly from his personal strength and stamina.
The clashes lasted longer as he learned how to fend off Brom. Now, when they went to sleep, Eragon was not the only one with bruises. (20.73)
Yes, young grasshopper! You have grown stronger! Oh, sorry, we were having a Kung Fuflashback. Much like Caine in that old TV series, Eragon is gaining strength thanks to his master's instruction. Soon, he will "snatch the pebble" from Brom's hand. This makes a lot more sense if you watch the show (which you should—trust us).
Brom tossed what remained of his stick into the fire and said, "We're done with these; throw yours in as well. You have learned well, but we have gone as far as we can with the branches. […] It is time for you to use the blade." (21.70)
Oh, snap. Blade? Who said anything about a blade? Here Eragon has graduated from whack-a-stick to full-on, blade vs. blade action. Brom is upping the ante as Eragon gets stronger. It seems that, as he makes progress, the bar continues to shift higher and higher for him. Of course, how else would he improve?
The long days and strenuous work stripped Eragon's body of excess fat. His arms became corded, and his tanned skin rippled with lean muscles. (23.15)
Somebody's getting ready for beach season. Seriously, though, more than just a killer bod, Eragon's development is a sign of his progress as a Rider. Now he can ride a dragon, cast magic, and grate cheese on his super-cut abdominal muscles. Rock on, E!
I didn't know I could stay on while you did that without being strapped into the saddle, he said, grinning fiercely. (29.18)
Remember when a ride on Saphira's back left Eragon a quivering mess of scabs and tears? Like his other skills with the blade and with magic, his ability to ride on Saphira's back grows and improves. Look, ma! No hands! His comfort on this back of a twisting, twirling dragon shows just how much he's developed as a Rider.
Eragon slowly lowered his arm and backed away. It was the first time he had bested Brom without resorting to trickery. Brom picked up his sword […] "We're done for today." (31.32)
Eragon wins! Without even cheating! His victory in the sparring session with Brom signifies a whole new level of achievement for our hero. Brom recognizes this, too. While it might seem like sour grapes that he quits, it's actually because he realizes that he has nothing left to teach Eragon about sword fighting. The dude has learned it all.
"But you can't indulge in wanton violence. Where is your empathy?" growled Eragon, pointing at the head. (47.39)
The head here belongs to the slaver Torkenbrand. Murtagh removed it from his body, although Torkenbrand was defenseless at the time. Murtagh sees this is a totally justified act, but Eragon is on the other side of the fence. His point is that, justified or not, raw power without empathy (an understanding of one's fellow man) is not just. This drives a wedge between Eragon and his fighting buddy. Whose position makes more sense to you?