I would split this up into two main reasons; the first being the need for experienced leaders in battle for strategy and morale, the second being the social expectations stressed as important during the medieval period.

The first point is largely self explanatory, while an army may be made up with some experienced fighters of lower classes, such as sergeants, in order for the force to work effectively there needed to be a strong sense of leadership in the form of a noble who was viewed high in society.

The danger to the nobles would not actually be too great, with Agincourt being a notable exception, as pitched-battles were avoided by both armies during warfare. As a pitched-battle lead to large losses on both sides they only occurred when one side believed it had a huge advantage over the other, and as such they were uncommon. Instead wars were largely fought with sieges, with defense often being done by positioning the defending force close enough to the siege that the attackers had to raise the siege and move to a safer location due to the threat posed upon them. This tactics was also used outside of sieges to force opposing armies away by shadowing them, often relying on the opponents being aware of your presence but not the exact location. Since this was a much safer way to wage war the nobility was under less threat than they would be in pitched-battle.

Building upon this is relationship between a liege and their vassal, with the expectations for a vassal to perform direct service to their liege in a demonstration of loyalty, as well as to gain reputation and standing. I would go further into that topic but I'm a little too busy to dig out my books on the topic to serve it justice.

On the other hand there were also expectations of a strong ruler, especially with the warrior nobility of the medieval period. I would say there are two main schools of thoughts regarding the social expectations of the warrior nobility, the first being "The Noble Habitus" as proposed by Crouch and the second being the Code of Chivalry as proposed by Gautier, with the Noble Habitus being largely accepted as the predecessor to chivalry. Both of these sets of expectations stress the importance of military action; in the Noble Habitus the importance of hardihood and bravery are stressed (which are to be demonstrated in battle) and loyalty (loyalty to your liege was expected to be demonstrated in battle). While the code of chivalry stresses the importance of making war with infidels without hesitation and the expectation to perform all feudal duties (which includes aiding your liege in battle). If a noble avoided fighting they would be looked down upon and seen as weak for not conforming to the expectations of society. The importance of social standing can not be stressed enough as it was required to be successful and keep control of your possessions, you could not be seen as weak and so you had to do what was expected of you.

So in conclusion the nobility fought in battle because, while it was dangerous, it was not as dangerous as one might expect, and because it was required for them to maintain their social standing and through it their possessions.

A. Musson, Expectations of the Law in the Middle Ages (Boydell & Brewer, 2001)

For understanding the relationship between liege and vassal.

M. Keen, Medieval Warfare: A History (Oxford, 1999)

For understanding how battles were fought in the period.

D. Crouch, The Birth of the Nobility (Routledge, 2005)

In this book Crouch proposed the Noble Habitus as the set of expectations of the warrior nobility which later evolved into Chivalry.

L. Gautier, Chivalry, Translated by D.C. Dunning (London, 1965)

One of the, if not the, most important work on Chivalry, it is largely outdated by modern work however Gautier's proposed code of Chivalry is largely considered to be accurate.

Edit: Formatting and explanation of the sources.

MrTorson already provided a really good answer but there are a few things I'd like to follow-up on.

Perhaps the main reason why nobles took a prominent role in battle in the Middle-Ages was because they had to, at least if they wanted to win. In 'The Age of Cavalry revisited' Stephen Morillo examined how the ability to field armies of well-trained, cohesive infantry fell away in the Middle Ages. In the absence of large, professional forces heavy cavalry became the vital core of most armies. This is where the nobles come in. Nobles didn't really serve 'in the ranks' as such, at least not how we would think of it today. A lord would lead his retinue of men-arms to battle, these men at arms would usually be nobility themselves just not first-born sons who stood to inherit land. The lord and his retinue made up the bodies of heavy cavalry around which medieval armies were based. As warrior-aristocrats these men were better fed, better trained and better organised than the rest of a medieval army and, as a result of being kept together on a permanent basis during which they would take part in training and tournaments together, these bodies developed much greater cohesion than the majority of footmen who made up the numbers in medieval armies who would have been levied at the start of a campaign and would likely never have met the men fighting beside them previously. In 'The Battle of Crecy' Andrew Ayton examines the importance of the experienced, cohesive cores of English retinues to their ultimate success.

It's also worth noting that in the majority of battles in the Middle Ages knights would have been protected by the code of chivalry as well as the ransom they could bring to their captor as long as they were kept alive. If they were killed, not so much (note: chivalry and ransoms would not apply to non-nobles in armies who would face much less certain fates). iirc at Agincourt King Henry only decided to execute his noble prisoners due to the fact that there were so many of them and so few English soldiers, so that if the French launched another attack the prisoners would likely be able to break free and arm themselves. Even then the English knights refused and it was ultimately the longbowmen (outside the protection of chivalry) who performed the executions.