Short answer:

Men hungry for cash and/or glory, quite often. Other times, men compelled by someone behind them.

Discussion:

It was more dangerous, but it came with more glory and indeed often more cash and booty. We have records of commanders from the ancient right up to the modern promising -- and delivering-- big rewards, to "the first man on the wall/into the breach".

People have reflected on the tension between the dreams of glory of young men and the reality of war for centuries

Laurence Sterne reflected on these ideas in his 1759 novel, Tristram Shandy

’Tis one thing, brother Shandy, for a soldier to hazard his own life—to leap first down into the trench, where he is sure to be cut in pieces:——’Tis one thing, from public spirit and a thirst of glory, to enter the breach the first man,—to stand in the foremost rank, and march bravely on with drums and trumpets, and colours flying about his ears:——’Tis one thing, I say, brother Shandy, to do this,—and ’tis another thing to reflect on the miseries of war;

. . . but Sterne was more philosophical than most young soldiers. We've got plenty of evidence that glory, cash and compulsion were all big parts in getting soldiers to do dangerous things.

Here's part of an order issued by General "Mad Anthony" Wayne

[T]he General cannot have the least doubt of a Glorious Victory; and he hereby most Solemnly engages to Reward the first man who enters the works with Five Hundred Dollars, and will represent the conduct of every Officer and Soldier who distinguishes himself on this occasion in the most favorable point of view to His Excellency, whose greatest pleasure is in rewarding merit.

[cited in R.J. Wood, "Leaders and Battles: The Art of Military Leadership" [Presidio Press:1995]

. . . such measures were typical. There were medals, prestige and cash rewards throughout history to men who distinguished themselves in dangerous ways, that's part of the dynamic of morale that commanders managed.

Relatively low ranking men became famous through the Army and to their commanders through this kind of bravery. Napoleon's Lieutenant Legros justified his sobriquet l'Enfonceur ("the Basher") with actions at Waterloo, where apparently went to work on the gates at Hougoumont with a sapper's axe, breaking open the gates-- he has the glory, but he gets killed inside along with most of his men.

Caesar mentions such bravery, for example in the Gallic Wars:

Lucius Fabius, a centurion of the VIIIth Legion, who was known to have said that day among his company that he was spurred on by the rewards at Avaricum, and would allow no one to mount the wall before him, got three men of his company, was lifted up by them, and mounted the wall. Then he in turn took hold of them one by one and pulled them up on to the wall.

That Caesar sees fit to mention a centurion by name demonstrates the prestige that attaches to such an act-- that presumably made Lucius Fabius a celebrated name among other soldiers, in a military culture that's a very big incentive.

One might also observe that the position of men outside the walls under fire might not be enviable either -- that's a very exposed position. Being "first on the wall" was riskier, but not certain death, and it came with rewards which were sufficiently enticing to many men. One has to be wary of misrepresentation by commanders in such stories, though-- the incentive to characterize a now deceased soldier as a hero who leapt onto the wall, rather than was compelled to do so by the man behind him, you can see how such a spin might occur.