Skooba has provided a good answer. This is only to complement the details he has missed imo.
Tyrion did not need their protection as he was in Lannister army
False. Tyrion did need their protection. He did not trust his father's men and valued having his own men around him, whose loyalties would be directly to him, not to Casterly Rock, his father, his sister or his brother. Which is one of the reasons why Tyrion refused to hand over his band of clansmen to his father when he was appointed to the position of Hand of the King in his father's stead. In Chapter 69 of AGOT:
When they were alone, Lord Tywin glanced at Tyrion. “Your savages
might relish a bit of rapine. Tell them they may ride with Vargo Hoat
and plunder as they like-goods, stock, women, they may take what they
want and burn the rest.”
“Telling Shagga and Timett how to pillage is like telling a rooster
how to crow,” Tyrion commented, “but I should prefer to keep them with
me.” Uncouth and unruly they might be, yet the wildlings were his, and
he trusted them more than any of his father’s men. He was not about to
hand them over.
Tyrion never made a promise of the sort that he would send Armies of Westerland to wrest the Vale from Arryns and hand it over to Clansmen. He pledged to provide them weapons (Which he did) and they wrecked havoc in the Vale afterwards, as they knew themselves that the fight to carve a better future for themselves was theirs alone.
So he took the Clansmen along with him to KL.
Clansmen were fierce and honorable in their own way. They were also loyal to him which made them such valued commodity in Tyrion's eyes and he relied on them with many tasks of sensitive nature. They shouted "Halfman" as their battle cry:
Shagga had an axe in either hand. He smashed them together and made
them ring. “Halfman!” he shouted. Other Stone Crows picked up the cry,
and the Black Ears and Moon Brothers as well. The Burned Men did not
shout, but they rattled their swords and spears. “Halfman! Halfman!
Other than that as Skooba noticed, they had better chances of plunder (As they plundered Stannis's baggage trains) and lived a better standard of life serving as men of a Lannister than they were accustomed to in their native mountains. They complained about smell, bad water and swindling populace but they still remained, why? For Money, personal loyalty and better prospects.
And then of course there were fancy city-women to have as well to which Shagga was so partial. From ACOK:
“Shagga wants this woman.”
“Shagga wants every whore in this city of whores,” complained Timett
son of Timett.
In conclusion, concurring with Skooba, following seem to be their reasons to stick along:
- Better Prospects of Plunder
- Personal Loyalty to Tyrion
- Hope to keep the Lannister support for Mountain clans open
- Better living standards