To answer this question, it's important to get inside Harold Godwinsson's head.

Harold wasn't in line to the throne through birthright. He was one of the sons of Duke Godwin of Wessex, a minor earl who had, under King Cnut, risen to become arguably the most powerful landholder in England. Godwin was more of a warrior than a noble, and raised his sons accordingly - this led to Harold, and his brothers, being highly competent generals. Harold inherited the Duchy of Wessex, while his brothers held other titles - around three-quarters of England was under the Godwinssons by 1066.

Godwin had been a better rival to the previous King of England, Edward the Confessor. Indeed, at one point, Godwin had been banished - only to return with a fleet, and strong-arm Edward into restoring his titles, and indeed into giving him even more power than he had previously held. Meanwhile, Edward never had a son - either because he was celibate, or infertile, we're not sure - and wanted to make sure that England did not fall into Godwin's hands after his death. Therefore, he promised the English throne to his cousin William the Bastard of Normandy, who he had grown up with while the Danes ruled England.

Towards the end of Edward's life, he began ailing, and Harold was able to persuade him to name him, not William, as his heir - or so he would claim. Harold then went on an expedition to Normandy, where he allegedly swore to support William's claim on the throne. Why would he do this? The Bayeux Tapestry, made by Anglo-Saxon embroiderers, shows Harold leaving Willaim's court immediately after taking this oath - implying that he had been a prisoner, and the oath was the price of his freedom. Some scholars have speculated that Harold had gone to Normandy with an offer - if William would support Harold's claim, Harold would support William in his struggles against the King of France. We may never know for sure, however.

Then Edward dies in 1066, and immediately, Harold is coronated. England at the time technically used a system of elective succession, where the national council of nobles and clergymen, the Witan, would choose the new King. In practice, this had generally been simply confirming the son of the previous King's right to rule. However, in 1066, there was no clear candidate, and so they were presented with a choice. By blood, the most rightful candidate was Edgar Aethling, a fifteen-year old descendant of Aethelred the Unready. However, with two powerful foreign claimants to the throne, the Witan chose Harold Godwinsson, a powerful landholder and experienced general.

Immediately after taking the throne, Harold had raised an army and prepared to defend the south of England from a Norman invasion; while appointing his ally Morcar to the Duchy of Northumbria and having him handle the northern defences. The slighted former Duke of Northumbria, Harold's brother Tostig, attempted to attack from the Netherlands, but was easily rebuffed and forced to head all the way north to Scotland, which shows that the defences were fairly effective. However, no Norman invasion came, and soon Harold had to dismiss his armies, as keeping them standing was too expensive.

Then, a joint force of Harald Hadrada and Tostig Godwinsson attacked York, and defeated Morcar's forces there. Harold quickly raised his armies again, and led a forced march north to fight off the invasion. At Stamford Bridge in Yorkshire, Harold scored a decisive victory over the rival claimant and his slighted brother. His celebrations were cut short by news of a Norman landing in southern England - more specifically, in his own holdings of Wessex.

Harold then leads a forced march south, and quickly engages the Normans at Hastings.

Firstly, this wasn't a suicidal attempt, but in fact, would prove to be a close battle. In the initial stages of the battle, Harold looks set to win, with the Norman cavalry rebuffed time and again by the Anglo-Saxon heavy infantry, or huscarls. Unfortunately, the Anglo-Saxon men, likely over-confident after Stamford Bridge, broke ranks to chase down the 'fleeing' Normans, who then turned around and attacked those forces who had abandoned their shield wall formation. This was the turning point, and Harold's brothers Gyrth and Leofwine were killed, followed by Harold himself - at which point, the Anglo-Saxon army was leaderless, and broke.

Secondly, Harold was likely full of confidence, having won a hugely decisive victory against the Norse at Stamford Bridge. England had been constantly under fear of the Scandinavians since the reign of Aethelred the Unready, and to find such a decisive Anglo-Saxon victory over them, you'd have to go back to the early 10th century. That victory had also come directly from a forced march - why wouldn't Harold try the same tactic again?

Then you have the fact that Harold likely despised William. If William did indeed imprison Harold to demand an oath - as seems likely - that would put plenty of fuel in Harold's fire. Even without that, William was a foreigner invading his new kingdom, and while there, William was ravaging parts of Wessex that belonged directly to Harold himself. In those circumstances, Harold's motives for engaging in battle were clear.

Sources

For anything on the Anglo-Saxons, Higham & Ryan's 'The Anglo-Saxon World' is my holy bible.

The Bayeux Tapestry, the Anglo-Saxon Chronicle, and Orderic Vitalis are the primary sources most worth looking into.