There's no answer given in the movie1, but there are a few equally plausible explanations:
- He didn't read that far. When he reads the contract, Bilbo is more than mildly hung up on the liability clauses:
Bilbo: Terms: Cash on delivery, up to but not exceeding one fourteenth of total profit, if any. Seems fair. Eh, Present company shall not be liable for injuries inflicted by or sustained as a consequence thereof including but not limited to lacerations... evisceration... incineration?
Bofur: Oh, aye, he’ll melt the flesh off your bones in the blink of an eye.
Bilbo: Huh.
Balin: You all right, laddie?
Bilbo: Uh, yeah... Feel a bit faint.
An Unexpected Journey (2013)
It's possible that, after talking himself into being okay with being burned alive by a dragon, he just skipped to the bottom and clicked "I Agree"
He didn't expect it to be an issue. This is something people do all the time when entering into contracts; they assume something won't be an issue, so they don't make a fuss about it. Is this particularly wise? No, not remotely. But it's possible
He wasn't in his right mind. Bilbo's decision to join the Dwarves is shown as being a very spur-of-the-moment decision, not particularly well thought-out. Even if he did read the arbitration clause, it's possible he (in his temporary madness) simply didn't process how big of a deal it was.
Personally, I think the most likely answer is that he didn't read that term of the contract, or at least didn't internalize it; we have evidence from later in the series that Bilbo may not have payed particularly close attention to the contract he signed. A major plot point in the film (and also in the book) is that Bilbo takes the Arkenstone as the entirety of his share of the treasure:
Thranduil: The Heart of the Mountain. The King's Jewel.
Bard: And worth a king's ransom. How is this yours to give?
Bilbo: I took it as my fourteenth share of the treasure.
The Battle of the Five Armies (2014)
But, as lawyer James Daily notes in a post on his blog analyzing the full text of the contract2, this is blatantly a violation of one of the terms of that agreement:
Burglar acknowledges and agrees that each item of the Company's valuables, goods, money or merchandise which he recovers from the Lonely Mountain [the 'Recovered Goods'] during the term of his engagement with the Company, shall remain the Property of the Company at all times, and in all respects, without limitation.
Furthermore, the company shall retain any and all Recovered Goods until such a time as a full and final reckoning can be made, from which the Total Profits can then be established. Then, and only then, will the Burglar's fourteenth share be calculated and decided.
So, despite his claims, the Arkenstone was never actually his to give.
1 The contract as written doesn't appear in the book
2 Based on a full-sized, officially-licensed replica