Tolkien’s heroes come from a different culture. Call it an Elven culture. Even the Rohirrim are not given to boasting about themselves. There is a subtle understanding among the heroic characters that you just don’t prattle on and tell people how good you are. A man is judged by his deeds, not by his willingness to endure challenger after challenge, duel after duel. When a great hero, like Turin, puts a companion to shame (as he did Gwindor in the councils of Nargothrond), the deed is not remembered well.
The restraint we see in Aragorn, Eomer, and others is undoubtedly due to the fact that they have grown up in the shadow of the Elves. Elven culture undoubtedly imparted a set of values to the Edain and other peoples under their influence, but the Edain and their relatives (the Northmen) valued bravery and action. And, if anything, the Elves could be brave and active. Even though Eomer has probably never seen an Elf, he knows of them, has heard tales of them, and fears them or at least respects them. When Eomer threatens Gimli Legolas quickly draws and arrow, fits it to his bow, and aims it at Eomer. No one doubts the Elf can and will kill Eomer if he gives the order to his men to kill Gimli. And Legolas is never rebuked.
The Elves were not the manliest of men in Middle-earth. Hurin was probably the strongest hero on record. He slew seventy trolls and countless Orcs, yelling out “Aure entuluva! (Day shall come again!)” before he was finally overborne and taken prisoner. His son Turin survived disaster after disaster, walking away from more battles than many long-lived Elven lords had. And Beren faced the depths of Angband, robbing the Dark Lord himself.
These names mean little to the first-time reader of The Lord of the Rings but they mean a great deal to Aragorn. And Eomer undoubtedly knows something of the ancient heroes, too, since his grandmother came from Gondor. The deeds of the great heroes cast a long shadow in Middle-earth, and its difficult for succeeding generations to match them. Isildur, for example, broke into the well-guarded royal palace of Númenor to steal a sapling of the White Tree. That’s a pretty significant accomplishment, but does it rival Hurin’s stand at the fens of Serech, or Turin’s defeat of Glaurung, or Earendil’s defeat of Ancalagon (not to mention Earendil’s years of voyages to find Valinor)?