the ability to crush her enemies.

The men of the mead hall bring out all their weapons to help arm Beowulf. Unferth, who once thought Beowulf was a braggart, now lends him his mighty sword, Hrunting, to fight the monster. Beowulf primarily rises to this daunting challenge because, as Hrothgar later says, he values the higher principles of honor and protecting civilization more than his own life.

Grendel is not only annoyed by the noise at Heorot; he's also bothered by the merriment and joy that occurs in the famous mead hall. Grendel is a descendent of Cain, the famed Biblical outcast who murdered his brother and was condemned to wander the earth. Thus, Grendel is cut off from larger communities, and so the sounds of celebration and joy (which we can also see as representations of community, friendship, and belonging) anger him and drive him to attack the hall. As such, the universal conflict can be seen as not only the continuation of a Biblical conflict, but also as the conflict between an outcast and a community. Indeed, Grendel can be seen as an archetypal version of the outcast who tries to punish the community for exiling him and refusing him friendship. Viewed in this light, Grendel almost becomes a sympathetic creature, although he's pretty much represented as a vicious brute in the poem. If you're interested in seeing an alternate version of Grendel, John Gardner's novel Grendel is a good place to start.  

Grendel is annoyed by the noise of the building of Herot, King Hrothgar's home for his soldiers. Then, the noise of the soldiers celebrating in Herot wakes him up so he begins a series of attacks. Beowulf arrives to help Hrothgar because Hrothgar had ended a feud for Beowulf’s father, Edgetho,when it threatened to cause war between the Geats and the Wulfings (a Germanic tribe).
Grendel is regarded as the descendant of Cain, the first murderer whose story is told in the Old Testament, and the sword Beowulf uses to cut off the dead Grendel's head is decorated with depictions of the Old Testament’s giants who were destroyed by the flood. Thus, the question "Am I my brother's keeper?" is echoed in Beowulf's response to Grendel's attacks. Beowulf obviously feels indebted to Hrothgar and eagerly defends the king and his men against a murderer.Unlike Cain, who killed his brother, Beowulf defends his fellow man.

Grendel's mother kills Aeschere, Hrothgar's closest friend and advisor, as part of a blood feud. The she-monster seeks payment in kind for the death of her son, and this means that, according to time-honored tradition followed by warriors and monsters alike, Hrothgar must pay for Grendel's death with the life of his closest friend and advisor.

In other words, Grendel's mother has lost the person—or rather, monster—closest to her, and so, according to the blood feud tradition, Hrothgar must suffer the same fate in return. Beowulf is on good terms with Hrothgar, but he's not as close to him as Aeschere is. This means that, although he was responsible for slaying Grendel, he is not the one upon whom Grendel's mother will exact her terrible revenge.

In the first chapter of Beowulf, Grendel's reasoning behind his attacks on Heorot are explained.

Then an evil creature who dwelt in darkness, full of envy and anger, was tormented by the hall's jubilant revel day by day, as the harps resounded loud, and the song of the singer called out clearly.

Here, readers are made aware that Grendel is both angry and jealous (envy). His anger comes from the "jubilant revel" heard coming from Heorot. While not divulged as to why Grendel is angry and jealous yet, readers know that Grendel is not happy.

The singer sang with the knowledge of tales from man's primeval time: how the Almighty fashioned the earth—a radiant plain rimmed by water—and delighted in its splendor.

The reader is given more information about what Grendel is angry about (given his feelings about the singing). One can infer that Grendel does not like the singing about God and how he has given the people of earth all they need. It is not until the following section that readers come to realize where Grendel's jealousy and hatred stem from.

This accursed one had long dwelled with monsters since the Creator had decreed his exile. On the kin of Cain did the sovereign God avenge the slaughter of Abel; Cain gained nothing from this feud and was driven far from the sight of men for that slaughter. From him awoke all those dire breeds: ogres, elves, and phantoms that warred with God a lengthy while; He paid their wage to them!

Essentially, Grendel is not happy with the fact that he cannot live in the light of God because his ancestors have been exiled. Grendel, given his exile, is not allowed to revel in the light of the world--he cannot take part in celebrations of God in a place like Heorot.

In the end, Grendel's anger stems from the fact that he was born into exile (without having done anything wrong himself). His jealously stems from the fact that he cannot live the same life as those at Heorot.

Good question. There are many reasons. Revenge is one. He must destroy here because of her attempt to seek revenge on Beowulf and his men for killing her son. This is similar to the blood feuds the Anglo-Saxons fought.

Glory and to cement his reputation are also factors. Remember, Beowulf\'s people, the Celts and later the Anglo-Saxons before Augustine converted King Ethelbert to Christianity, didn\'t have a concept of afterlife. The only way to live on was to perform heroic deeds that would be recorded in song and poem by the bards and recited after you were dead. By not only killing the monster Grendel but also killing his mother, Beowulf\'s reputation would be even more incredible and sure to be retold.

Bravery is another factor. Remember the heroic code to which Beowulf adheres - loyalty, bravery, and generosity - are embodied in all of his battles. He is being loyal to Hrothgar by sailing to his land to battle these monsters. If he were to leave, he would not fulfill his obligation. He is also being brave by seeking her out and trying to kill her. Finally, if he destroys her, his men can share her treasure and weapons. All of these are vital to the Celtic and Anglo-Saxon ways of life.

All of these are great possible answers.  At this point in the poem, however, Beowulf is going mainly for justice/revenge.  The men had celebrated heartily after Grendel had been killed and his arm is hung in the rafters as a sort of trophy.  They let their guard down and once again return to the Hall of Herot as a safe haven and place to lie down to sleep without worries.  Grendel's mother comes without warning and not only takes back her son's arm, but kills a few men (among them Hrothgrar's best buddy).  Beowulf takes it upon himself to seek justice for harm done and revenge for Hrothgar (as he owes this to the long-family friend for once helping out Beowulf's father).

In Beowulf, the titular character is motivated by money and honor to kill Grendel’s mother.

Grendel’s mother is infuriated and distraught by her son’s death. In a blind rage, she storms into Heorot, retrieves her dead son’s severed arm, and kills Aeschere, who is Hrothgar’s closest friend and top advisor. After this, she retreats to her swamp lair.

Hrothgar is devastated by the death of his friend and promises to give Beowulf a large amount of gold in exchange for avenging Aeschere’s death by finding and killing Grendel’s mother.

Beowulf would probably have agreed to kill Grendel’s mother without the promise of a monetary reward. Beowulf is an honorable, brave, and loyal man. He enjoys challenges and jumps at any opportunity to be helpful and prove his worth as a warrior.

Beowulf feels he owes a debt to Hrothgar, who once saved his father’s life. He is eager to help the king when he learns that Grendel is terrorizing Heorot. He probably would have been happy to help Hrothgar and have an opportunity to strengthen his skills as a warrior by hunting and killing Grendel’s mother, even if he had not been offered gold as a reward.